utils-0.86/000755 000765 000024 00000000000 14621610515 015714 5ustar00christopherwensleystaff000000 000000 utils-0.86/PackageInfo.g000644 000765 000024 00000014150 14621444241 020235 0ustar00christopherwensleystaff000000 000000 ############################################################################# ## ## PackageInfo.g file for the package Utils ## SetPackageInfo( rec( PackageName := "utils", Subtitle := "Utility functions in GAP", Version := "0.86", Date := "16/05/2024", # dd/mm/yyyy format License := "GPL-2.0-or-later", Persons := [ rec( LastName := "Breuer", FirstNames := "Thomas", IsAuthor := true, IsMaintainer := false, Email := "sam@math.rwth-aachen.de", WWWHome := "https://www.math.rwth-aachen.de/~Thomas.Breuer", Place := "Aachen", Institution := "Lehrstuhl für Algebra und Zahlentheorie, RWTH Aachen" ), rec( LastName := "Gutsche", FirstNames := "Sebastian", IsAuthor := true, IsMaintainer := false, Email := "gutsche@mathematik.uni-siegen.de", WWWHome := "https://sebasguts.github.io/", Place := "Siegen", Institution := "University of Siegen" ), rec( LastName := "Horn", FirstNames := "Max", IsAuthor := true, IsMaintainer := false, Email := "mhorn@rptu.de", WWWHome := "https://github.com/mhorn", ), rec( LastName := "Hulpke", FirstNames := "Alexander", IsAuthor := true, IsMaintainer := false, Email := "hulpke@math.colostate.edu", WWWHome := "https://www.math.colostate.edu/~hulpke", ), rec( LastName := "García-Sánchez", FirstNames := "Pedro", IsAuthor := true, IsMaintainer := false, WWWHome := "http://www.ugr.es/local/pedro", Email := "pedro@ugr.es", ), rec( LastName := "Jefferson", FirstNames := "Christopher", IsAuthor := true, IsMaintainer := false, WWWHome := "https://caj.host.cs.st-andrews.ac.uk/", Email := "caj21@st-andrews.ac.uk", ), rec( LastName := "Kohl", FirstNames := "Stefan", IsAuthor := true, IsMaintainer := false, Email := "stefan@mcs.st-and.ac.uk", WWWHome := "https://www.gap-system.org/DevelopersPages/StefanKohl/" ), rec( LastName := "Lübeck", FirstNames := "Frank", IsAuthor := true, IsMaintainer := false, Email := "Frank.Luebeck@Math.RWTH-Aachen.De", WWWHome := "https://www.math.rwth-aachen.de/~Frank.Luebeck", ), rec( LastName := "Wensley", FirstNames := "Chris", IsAuthor := true, IsMaintainer := true, Email := "cdwensley.maths@btinternet.com", WWWHome := "https://github.com/cdwensley", Place := "Llanfairfechan", Institution := "" ) ], Status := "deposited", ## CommunicatedBy := " ", ## AcceptDate := " ", SourceRepository := rec( Type := "git", URL := "https://github.com/gap-packages/utils" ), IssueTrackerURL := "https://github.com/gap-packages/utils/issues", PackageWWWHome := "https://gap-packages.github.io/utils", README_URL := Concatenation( ~.PackageWWWHome, "/README.md" ), PackageInfoURL := Concatenation( ~.PackageWWWHome, "/PackageInfo.g" ), ArchiveURL := Concatenation( ~.SourceRepository.URL, "/releases/download/v", ~.Version, "/", ~.PackageName, "-", ~.Version ), ArchiveFormats := ".tar.gz .zip", AbstractHTML := "The Utils package provides a collection \ of utility functions gleaned from many packages.", PackageDoc := rec( BookName := "Utils", ArchiveURLSubset := ["doc"], HTMLStart := "doc/chap0_mj.html", PDFFile := "doc/manual.pdf", SixFile := "doc/manual.six", LongTitle := "Utility functions in GAP", Autoload := true ), Dependencies := rec( GAP := ">=4.10.1", NeededOtherPackages := [ ], SuggestedOtherPackages := [ [ "curlInterface", ">= 2.3.0" ] ], ExternalConditions := [ ] ), AvailabilityTest := ReturnTrue, Autoload := false, Keywords := [ ], BannerString := Concatenation( "Loading Utils ", String( ~.Version ), " for GAP 4.13.0", " - a collection of utility functions.\n" ), TestFile := "tst/testall.g", AutoDoc := rec( TitlePage := rec( Copyright := Concatenation( "©right; 2015-2024, The GAP Group.

\n", "The &Utils; package is free software; you can redistribute it ", "and/or modify it under the terms of the GNU General ", "Public License as published by the Free Software Foundation; ", "either version 2 of the License, or (at your option) ", "any later version.\n" ), Abstract := Concatenation( "The &Utils; package provides a space for utility functions ", "in a variety of ⪆ packages to be collected together ", "into a single package. In this way it is hoped that they will ", "become more visible to package authors.\n", "

\n", "Any package author who transfers a function to &Utils; ", "will become an author of &Utils;.\n", "

\n", "If deemed appropriate, functions may also be transferred ", "from the main library.\n", "

\n", "Bug reports, suggestions and comments are, of course, welcome.\n", "Please contact the last author at ", "cdwensley.maths@btinternet.com ", "or submit an issue at the GitHub repository ", "https://github.com/gap-packages/utils/issues/.\n" ), Acknowledgements := Concatenation( "This documentation was prepared using the ", "&GAPDoc; and ", "&AutoDoc; packages.

\n", "The procedure used to produce new releases uses the package ", "GitHubPagesForGAP ", " ", "and the package ReleaseTools.

" ), ) ), )); utils-0.86/README.md000644 000765 000024 00000003471 14621444241 017201 0ustar00christopherwensleystaff000000 000000 [![Build Status](https://github.com/gap-packages/utils/workflows/CI/badge.svg?branch=master)](https://github.com/gap-packages/utils/actions?query=workflow%3ACI+branch%3Amaster) [![Code Coverage](https://codecov.io/github/gap-packages/utils/coverage.svg?branch=master&token=)](https://codecov.io/gh/gap-packages/utils) # The GAP 4 package `Utils` ## Introduction This package collects together utility functions from a selection of GAP packages in order to make them more widely visible to other package authors. Other generally useful functions, which are not deemed suitable for the main library, and also welcome. For example, recent additions are functions to convert certain types of group to Magma format. ## Distribution * The GitHub package repository is at * and the latest release is available at ## Copyright The `Utils` package is Copyright {\copyright} The GAP Group, 2015-2024. `Utils` is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. For details, see ## Installation * unpack `utils-.tar.gz` in the `pkg` subdirectory of the GAP root directory. * From within GAP load the package with: gap> LoadPackage("utils"); * The file `manual.pdf` is in the `doc` subdirectory. ## Contact If you have a question relating to `Utils`, encounter any problems, or wish to suggest other functions to be transferred to the package, please * email * or report an issue on the GitHub issue tracker at: utils-0.86/makedoc.g000644 000765 000024 00000002261 14621444241 017471 0ustar00christopherwensleystaff000000 000000 ## makedoc.g for the package Utils ## This builds the documentation of the Utils package. ## Needs: GAPDoc & AutoDoc packages, latex, pdflatex, mkindex ## call this with GAP from within the package root directory ## LoadPackage( "GAPDoc" ); LoadPackage( "AutoDoc" ); AutoDoc( rec( scaffold := rec( ## MainPage := false, includes := [ "intro.xml", "print.xml", "lists.xml", "number.xml", "groups.xml", "matrix.xml", "iterator.xml", "record.xml", "download.xml", "others.xml", "obsolete.xml", "transfer.xml" ], bib := "bib.xml", entities := rec( AutoDoc := "AutoDoc", ResClasses := "ResClasses", RCWA := "RCWA", XMod := "XMod", Home := "Home", Magma := "{\\sf Magma}", MathJax := "{\\sf MathJax}" ) ) )); utils-0.86/lib/000755 000765 000024 00000000000 14621444241 016463 5ustar00christopherwensleystaff000000 000000 utils-0.86/CHANGES.md000644 000765 000024 00000016317 14621444241 017317 0ustar00christopherwensleystaff000000 000000 # CHANGES log for the 'Utils' package ## Version 0.86 for GAP 4.13.1 (16/05/24) * (16/05/24) there have been just two minor adjustments since version 0.85 ## Version 0.85 for GAP 4.12.2 (23/01/24) * (08/01/24) avoid trivial function wrappers in List and ForAll ## Version 0.84 for GAP 4.12.2 (11/09/23) * (11/09/23) changed manual and test for DirectSumDecompositionMatrices ## Version 0.83 for GAP 4.12.2 (29/06/23) * (29/06/23) added DirectSumDecompositionMatrices ## Version 0.82 for GAP 4.12.2 (09/02/23) * (23/12/22) changed email address, deleted institution ## Version 0.81 for GAP 4.12.1 (04/12/22) * (17/11/22) removed (the dead) pcp option from PcGroupToMagmaFormat so that the dependency on Polycyclic could be removed * (21/10/22) added OptionRecordWithDefaults in 7.2 from Christopher Jefferson * (04/10/22) declared PreImagesRepresentativeNC in init.g ## Version 0.77 for GAP 4.12.0 (25/09/22) * (25/09/22) added Download operation by Thomas Breuer - new Chapter 8 ## Version 0.76 for GAP 4.12.0 (06/08/22) * (06/08/22) added LeftCoset operations ## Version 0.74 for GAP 4.11.1 (09/07/22) * (07/07/22) Max Horn replaced BIND_GLOBAL with BindGlobal (transfer process) * (05/07/22) CentralProduct added by Thomas Breuer ## Version 0.72 for GAP 4.10.1 (16/11/21) * (06/04/21) Switch CI to use GitHub Actions: PR#37 ## Version 0.69 for GAP 4.10.2 (29/11/19) * (22/11/19) added DirectProductOfAutomorphismGroups * (19/11/19) added DirectProductOfFunctions ## Version 0.67 for GAP 4.10.2 (04/09/19) * (04/09/19) accepted PR34 - changed example in 6.2.1 * (25/08/19) fixed typos with UnorderedPairsIterator in manual ## Version 0.65 for GAP 4.10.2 (11/07/19) * (11/07/19) groups.tst fails in gapdev - temporary(?) fix ## Version 0.64 for GAP 4.10.1 (17/06/19) * (14/06/19) added Iterators chapter: AllSubgroupsIterator CartesianIterator, UnorderedPairsIterator ## Version 0.63 for GAP 4.10.1 (29/05/19) * (28/05/19) added IdempotentEndomorphisms, IdempotentEndomorphismsData * (17/05/19) + AllIsomorphismsIterator, AllIsomorphismsNumber, AllIsomorphisms * (16/02/19) added License field in PackageInfo.g ## Version 0.61 for GAP 4.10.0 (28/11/18) * (28/11/18) made ExponentOfPrime obsolete * (27/11/18) made several AutoDoc functions obsolete (they remain in AutoDoc): FindMatchingFiles; CreateDirIfMissing; StringDotSuffix; SetIfMissing ## Version 0.59 for GAP 4.9.3 (04/10/18) * (04/10/18) made PrintApplicableMethod obsolete ## Version 0.58 for GAP 4.9.3 (12/09/18) * (08/09/18) marked PrintOneItemPerLine as obsolete: use Perform(L,Display); * (07/09/18) marked ExponentOfPrime as obsolete: use PValuation * (05/09/18) transferred code in combinat.g{d,i} to end of lists.g{d,i} * (04/09/18) removed all "if OKtoReadFomUtils( "RCWA" / "ResClasses" )" ## Version 0.57 for GAP 4.9.1 (02/06/18) * (02/06/18) new PrintSelection example due to diff with all packages loaded removed XMod from list of packages to be loaded * (27/03/18) revised file tst/testing.g ## Version 0.54 for GAP 4.9.0 (12/02/18) * (22/01/18) PrintOneItemPerLine plus alternative method for iterators; added function PrintSelection(L,first,step,last) for lists/iterators * (06/01/18) rebuilt the manual using the AutoDoc package * (22/12/17) removed examples/ folder ## Version 0.49 for GAP 4.8.8 (05/12/17) * (05/12/17) removed QPA functions PositionsNonzero and NullList * (21/11/17) changed record.tst so that 4r8 and dev give the same result ## Version 0.48 for GAP 4.8.8 (14/09/17) * (14/09/17) fixes to the gh-pages folder so that README.md is displayed * (12/09/17) corrected web addresses for Stefan and Frank * (04/07/17) README and CHANGES converted to README.md and CHANGES.md ## Version 0.46 for GAP 4.8.6 (08/02/17) * (08/02/17) added Polycyclic as a needed package * (07/02/17) added code for converting matrix groups to Magma strings * (03/02/17) added code for converting perm- and pc-groups to Magma strings * (02/02/17) copied `gaplog.css` to `utils/doc/` from RCWA archive ## Version 0.44 for GAP 4.8.6 (17/01/17) * (08/12/16) added PositionsNonzero and NullList from QPA * (05/12/16) added `tst/loadall.g` and adjusted `tst/testall.g`, `tst/*.tst` * (14/11/16) issue #2 reopened by Max: EpimorphismByGeneratorsNC removed ## Version 0.43 for GAP 4.8.5 (20/10/16) * (18/10/16) now using bibliography file `bib.xml` of type `bibxmlext.dtd` ## Version 0.41 for GAP 4.8.3 (25/05/16) * (25/05/16) fixed issue #8 : `git rm doc/appendix.xml` * (17/03/16) added function PrintApplicableMethod ## Version 0.40 for GAP 4.8.3 (17/03/16) * (17/03/16) moved transfer procedure to a new chapter in the manual * (15/03/16) added fix to QuotientList requested by Stefan added many manual examples supplied by Stefan ## Version 0.39 for GAP 4.8.2 (04/03/16) * (04/03/16) corrected repeated "//" in `PackageInfo.g` * (03/03/16) updated transfers from RCWA and ResClasses (Stefan's input) * (26/02/16) updated README to show the release URL * (25/02/16) New release to GitHub using ReleaseTools (GitHubPagesForGAP) `makedocrel.g` renamed `makedoc.g` * (17/02/16) Removed date/version information from file headers. * (16/02/16) Moved PackageWWWHome to GitHub; added OKtoReadFromUtilsSpec to deal with fns taken out early * (12/02/16) Added warning to manual about the need to load packages. * (11/02/16) BindInRecordIfMissing -> SetIfMissing (from AutoDoc) moved a number of functions to `pending.g{d,i}` * (10/02/16) adjustments due to new RCWA 4.0.0 and ResClasses 4.1.1 ## Version 0.22 for GAP 4.8.1 (04/02/16) * (04/02/16) repository moved to gap-packages ## Version 0.21 for GAP 4.8.1 (03/02/16) * (03/02/16) added several more functions from RCWA * (29/01/16) switching to a protocol based on Frank's suggestion: if TestPackageAvailability("Bla", ">=x.(y+1)") = fail then ... ## Version 0.17 for GAP 4.8.1 (12/01/16) * (12/01/16) more edits to `oper.g` and `global.g` suggested by Chris J. ## Version 0.15 for GAP 4.8.0 (18/12/15) * (17/12/15) added more documentation to `global.g` and `oper.g` ## Version 0.14 for GAP 4.8.0 (16/12/15) * (16/12/15) changed global lists in oper.g to GLOBAL_REDECLARATION_LIST, GLOBAL_REDECLARATION_COUNT and GLOBAL_REINSTALLATION_LIST; added functions (in `oper.g`) AllowGlobalRedeclaration and AllowGlobalReinstallation; added GLOBAL_REBINDING_LIST and AllowGlobalRebinding (in `global.g`) to cope with BIND_GLOBAL. ## Version 0.13 for GAP 4.8.0 (15/12/15) * (15/12/15) added some functions from AutoDoc; packed up version 0.13 * (14/12/15) library file `oper.g` edited to prevent repeat declarations etc. * (14/12/15) `names.g{d,i}` now `first.g{d,i}` with `last.gd` added, containing UTILS_FUNCTION_NAMES and UTILS_FUNCTION_OPERS ## Version 0.11 for GAP 4.8.0 (30/11/15) * (30/11/15) added tests and documentation for ResClasses functions * (25/11/15) added `names.g{d,i}`, `tests.g{d,i}` * (25/11/15) added `sebastian.gi` to `lib/` * (24/11/15) added `rwca.g{d,i}` and `resclasses.g{d,i}` to `lib/` * (24/11/15) added `combinat.g{d,i}` to `lib/` utils-0.86/doc/000755 000765 000024 00000000000 14621610515 016461 5ustar00christopherwensleystaff000000 000000 utils-0.86/tst/000755 000765 000024 00000000000 14621444241 016527 5ustar00christopherwensleystaff000000 000000 utils-0.86/init.g000644 000765 000024 00000002373 14621444241 017035 0ustar00christopherwensleystaff000000 000000 ############################################################################# ## #W init.g GAP package `Utils' Chris Wensley ## #Y Copyright (C) 2015-2023, The GAP Group, ## if not IsBound( PreImagesRepresentativeNC ) then BindGlobal( "PreImagesRepresentativeNC", PreImagesRepresentative ); fi; if not IsBound( IsMatrixOrMatrixObj ) then BindGlobal( "IsMatrixOrMatrixObj", IsMatrix ); fi; if not IsBound( MutableCopyMatrix ) then BindGlobal( "MutableCopyMatrix", ShallowCopy ); fi; ## read the function declarations ReadPackage( "utils", "lib/start.gd" ); ReadPackage( "utils", "lib/files.gd" ); ReadPackage( "utils", "lib/groups.gd" ); ReadPackage( "utils", "lib/iterator.gd" ); ReadPackage( "utils", "lib/latex.gd" ); ReadPackage( "utils", "lib/lcset.gd" ); ReadPackage( "utils", "lib/lists.gd" ); ReadPackage( "utils", "lib/magma.gd" ); ReadPackage( "utils", "lib/maps.gd" ); ReadPackage( "utils", "lib/matrix.gd" ); ReadPackage( "utils", "lib/number.gd" ); ReadPackage( "utils", "lib/print.gd" ); ReadPackage( "utils", "lib/record.gd" ); ReadPackage( "utils", "lib/string.gd" ); if not IsBound( Download ) then ReadPackage( "utils", "lib/download.gd" ); BindGlobal("DOWNLOAD_FROM_UTILS", true); fi; utils-0.86/LICENSE.txt000644 000765 000024 00000043103 14621444241 017541 0ustar00christopherwensleystaff000000 000000 GNU GENERAL PUBLIC LICENSE Version 2, June 1991 Copyright (C) 1989, 1991 Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. 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It is safest to attach them to the start of each source file to most effectively convey the exclusion of warranty; and each file should have at least the "copyright" line and a pointer to where the full notice is found. Copyright (C) This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. Also add information on how to contact you by electronic and paper mail. If the program is interactive, make it output a short notice like this when it starts in an interactive mode: Gnomovision version 69, Copyright (C) year name of author Gnomovision comes with ABSOLUTELY NO WARRANTY; for details type `show w'. This is free software, and you are welcome to redistribute it under certain conditions; type `show c' for details. The hypothetical commands `show w' and `show c' should show the appropriate parts of the General Public License. Of course, the commands you use may be called something other than `show w' and `show c'; they could even be mouse-clicks or menu items--whatever suits your program. You should also get your employer (if you work as a programmer) or your school, if any, to sign a "copyright disclaimer" for the program, if necessary. Here is a sample; alter the names: Yoyodyne, Inc., hereby disclaims all copyright interest in the program `Gnomovision' (which makes passes at compilers) written by James Hacker. , 1 April 1989 Ty Coon, President of Vice This General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Lesser General Public License instead of this License. utils-0.86/read.g000644 000765 000024 00000001526 14621444241 017004 0ustar00christopherwensleystaff000000 000000 ############################################################################# ## #W read.g The Utils package Chris Wensley ## #Y Copyright (C) 2015-2023, The GAP Group, ## ## read the actual code ReadPackage( "utils", "lib/files.gi" ); ReadPackage( "utils", "lib/groups.gi" ); ReadPackage( "utils", "lib/iterator.gi" ); ReadPackage( "utils", "lib/latex.gi" ); ReadPackage( "utils", "lib/lcset.gi" ); ReadPackage( "utils", "lib/lists.gi" ); ReadPackage( "utils", "lib/magma.gi" ); ReadPackage( "utils", "lib/maps.gi" ); ReadPackage( "utils", "lib/matrix.gi" ); ReadPackage( "utils", "lib/number.gi" ); ReadPackage( "utils", "lib/print.gi" ); ReadPackage( "utils", "lib/record.gi" ); ReadPackage( "utils", "lib/string.gi" ); if IsBound(DOWNLOAD_FROM_UTILS) then ReadPackage( "utils", "lib/download.gi" ); fi; utils-0.86/tst/lists.tst000644 000765 000024 00000007676 14621444241 020441 0ustar00christopherwensleystaff000000 000000 #@local C, c4, d16, gens, i, J, K, L, n, P, trans ############################################################################## ## #W lists.tst Utils Package ## #Y Copyright (C) 2015-2022, The GAP Group ## gap> ReadPackage( "utils", "tst/loadall.g" );; gap> UtilsLoadingComplete; true ## SubSection 3.1.1 gap> List( [1..12], n->n^3 ); [ 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 ] gap> DifferencesList( last ); [ 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397 ] gap> DifferencesList( last ); [ 12, 18, 24, 30, 36, 42, 48, 54, 60, 66 ] gap> DifferencesList( last ); [ 6, 6, 6, 6, 6, 6, 6, 6, 6 ] gap> L := [ 0 ];; gap> DifferencesList( L ); [ ] gap> L := [ ];; gap> DifferencesList( L ); [ ] ## SubSection 3.1.2 gap> List( [0..10], n -> Factorial(n) ); [ 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 ] gap> QuotientsList( last ); [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ] gap> L := [ 1, 3, 5, -1, -3, -5 ];; gap> QuotientsList( L ); [ 3, 5/3, -1/5, 3, 5/3 ] gap> FloatQuotientsList( L ); [ 3., 1.66667, -0.2, 3., 1.66667 ] gap> QuotientsList( [ 2, 1, 0, -1, -2 ] ); [ 1/2, 0, fail, 2 ] gap> FloatQuotientsList( [1..10] ); [ 2., 1.5, 1.33333, 1.25, 1.2, 1.16667, 1.14286, 1.125, 1.11111 ] gap> Product( last ); 10. ## SubSection 3.1.3 gap> L := [1..20];; L[1]:=13;; gap> for i in [1..19] do > if IsOddInt(L[i]) then L[i+1]:=3*L[i]+1; else L[i+1]:=L[i]/2; fi; > od; gap> L; [ 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4 ] gap> SearchCycle( L ); [ 1, 4, 2 ] gap> n := 1;; L := [n];; gap> for i in [1..100] do n:=(n^2+1) mod 1093; Add(L,n); od; gap> L; [ 1, 2, 5, 26, 677, 363, 610, 481, 739, 715, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004 ] gap> C := SearchCycle( L ); [ 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754 ] gap> P := Positions( L, 157 ); [ 14, 26, 38, 50, 62, 74, 86, 98 ] gap> Length( C ); DifferencesList( P ); 12 [ 12, 12, 12, 12, 12, 12, 12 ] ## SubSection 3.1.4 ## this manual example not tested as answers vary from run to run ## gap> RandomCombination( [1..49], 6 ); ## [ 3, 5, 21, 24, 27, 31 ] ## SubSection 3.2.1 gap> J := [ [1,2,3], [3,4], [3,4], [1,2,4] ];; gap> DistinctRepresentatives( J ); [ 1, 3, 4, 2 ] gap> K := [ [3,4], [1,2], [2,3], [2,3,4] ];; gap> CommonRepresentatives( J, K ); [ [ 3, 3, 3, 1 ], [ 1, 3, 4, 2 ] ] gap> d16 := DihedralGroup( IsPermGroup, 16 ); SetName( d16, "d16" ); Group([ (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) ]) gap> c4 := Subgroup( d16, [ d16.1^2 ] ); SetName( c4, "c4" ); Group([ (1,3,5,7)(2,4,6,8) ]) gap> RightCosets( d16, c4 ); [ RightCoset(c4,()), RightCoset(c4,(2,8)(3,7)(4,6)), RightCoset(c4,(1,8,7,6,5, 4,3,2)), RightCoset(c4,(1,8)(2,7)(3,6)(4,5)) ] gap> trans := CommonTransversal( d16, c4 ); [ (), (2,8)(3,7)(4,6), (1,2,3,4,5,6,7,8), (1,2)(3,8)(4,7)(5,6) ] gap> IsCommonTransversal( d16, c4, trans ); true ## SubSection 3.3.1 gap> gens := GeneratorsOfGroup( DihedralGroup(12) ); [ f1, f2, f3 ] gap> String( gens ); "[ f1, f2, f3 ]" gap> BlankFreeString( gens ); "[f1,f2,f3]" ############################################################################# ## #E lists.tst . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/tst/print.tst000644 000765 000024 00000002027 14621444241 020420 0ustar00christopherwensleystaff000000 000000 #@local L, s5 ############################################################################## ## #W print.tst Utils Package ## #Y Copyright (C) 2015-2022, The GAP Group ## gap> ReadPackage( "utils", "tst/loadall.g" );; gap> UtilsLoadingComplete; true ## SubSection 2.1.1 gap> L := List( [1..20], n -> n^5 );; gap> PrintSelection( L, [18..20] ); 18 : 1889568 19 : 2476099 20 : 3200000 gap> PrintSelection( L, 2, 9 ); 2 : 32 11 : 161051 20 : 3200000 gap> PrintSelection( L, 2, 3, 11 ); 2 : 32 5 : 3125 8 : 32768 11 : 161051 gap> s5 := SymmetricGroup( 5 );; gap> PrintSelection( s5, [30,31,100,101] ); 30 : (1,5)(3,4) 31 : (1,5,2) 100 : (1,4,3) 101 : (1,4)(3,5) gap> PrintSelection( s5, 1, 30 ); 1 : () 31 : (1,5,2) 61 : (1,2,3) 91 : (1,3,5,2,4) gap> PrintSelection( s5, 9, 11, 43 ); 9 : (2,5,3) 20 : (2,4) 31 : (1,5,2) 42 : (1,5,2,3,4) ############################################################################# ## #E print.tst . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/tst/iterator.tst000644 000765 000024 00000005250 14621444241 021116 0ustar00christopherwensleystaff000000 000000 #@local c3c3, cart, G, h, it1, it2, iter, iter0, iter4, iterL #@local L, n, pairs0, pairs4, pairsL, s3, s4 ############################################################################## ## #W iterator.tst Utils Package ## #Y Copyright (C) 2015-2022, The GAP Group ## gap> ReadPackage( "utils", "tst/loadall.g" );; gap> UtilsLoadingComplete; true ## SubSection 7.1.1 gap> G := SmallGroup( 6,1);; gap> s3 := Group( (5,6), (6,7) );; gap> iter := AllIsomorphismsIterator( G, s3 );; gap> NextIterator( iter ); [ f1, f2 ] -> [ (6,7), (5,6,7) ] gap> n := AllIsomorphismsNumber( G, s3 ); 6 gap> AllIsomorphisms( G, s3 ); [ [ f1, f2 ] -> [ (6,7), (5,6,7) ], [ f1, f2 ] -> [ (5,7), (5,6,7) ], [ f1, f2 ] -> [ (5,6), (5,7,6) ], [ f1, f2 ] -> [ (6,7), (5,7,6) ], [ f1, f2 ] -> [ (5,7), (5,7,6) ], [ f1, f2 ] -> [ (5,6), (5,6,7) ] ] gap> iter := AllIsomorphismsIterator( G, s3 );; gap> for h in iter do Print( ImageElm( h, G.1 ) = (6,7), ", " ); od; true, false, false, true, false, false, gap> s4 := Group( (1,2),(2,3),(3,4) );; gap> AllIsomorphismsNumber( s3, s4 ); 0 gap> AllIsomorphisms( s3, s4 ); [ ] ## SubSection 7.1.2 gap> c3c3 := Group( (1,2,3), (4,5,6) );; gap> iter := AllSubgroupsIterator( c3c3 ); gap> while not IsDoneIterator(iter) do Print(NextIterator(iter),"\n"); od; Group( () ) Group( [ (4,5,6) ] ) Group( [ (1,2,3) ] ) Group( [ (1,2,3)(4,5,6) ] ) Group( [ (1,3,2)(4,5,6) ] ) Group( [ (4,5,6), (1,2,3) ] ) ## Subsection 7.2.1 gap> it1 := Iterator( [ 1, 2, 3 ] );; gap> it2 := Iterator( [ 4, 5, 6 ] );; gap> iter := CartesianIterator( it1, it2 );; gap> while not IsDoneIterator(iter) do Print(NextIterator(iter),"\n"); od; [ 1, 4 ] [ 1, 5 ] [ 1, 6 ] [ 2, 4 ] [ 2, 5 ] [ 2, 6 ] [ 3, 4 ] [ 3, 5 ] [ 3, 6 ] gap> ## try some trivial cases gap> iter0 := IteratorList( [ ] );; gap> iter4 := IteratorList( [ 4 ] );; gap> cart := CartesianIterator( iter0, iter0 );; gap> IsDoneIterator( cart ); true gap> cart := CartesianIterator( iter0, iter4 );; gap> IsDoneIterator( cart ); true gap> cart := CartesianIterator( iter4, iter0 );; gap> IsDoneIterator( cart ); true ## Subsection 6.2.2 gap> L := [6,7,8,9];; gap> iterL := IteratorList( L );; gap> pairsL := UnorderedPairsIterator( iterL );; gap> while not IsDoneIterator(pairsL) do Print(NextIterator(pairsL),"\n"); od; [ 6, 6 ] [ 6, 7 ] [ 7, 7 ] [ 6, 8 ] [ 7, 8 ] [ 8, 8 ] [ 6, 9 ] [ 7, 9 ] [ 8, 9 ] [ 9, 9 ] gap> iter4 := IteratorList( [ 4 ] ); gap> pairs4 := UnorderedPairsIterator(iter4); gap> NextIterator( pairs4 ); [ 4, 4 ] gap> IsDoneIterator( pairs4 ); true gap> pairs0 := UnorderedPairsIterator( iter0 );; gap> IsDoneIterator( pairs0 ); true utils-0.86/tst/groups.tst000644 000765 000024 00000021266 14621444241 020611 0ustar00christopherwensleystaff000000 000000 #@local A, a, ac9, aq8, b, c, c1, c2, c2c6, c3, c4, c4c3, c6, c9 #@local cp1, cp2, cp3, d, D12, d8, data, dp, epi, f, f1, f2, G, g #@local g1, g2, g3, genA, gens, i, idcopy, idem, idemc2, idemim, idemim2 #@local im2, image2, images, imi, info, info2, info3, lc1, lc2, lc3, lc4 #@local len, lfs, ok, Pfi, phi, q8, rc1, rc4, s3, S4, s4, UFS, ufs ############################################################################## ## #W groups.tst Utils Package ## #Y Copyright (C) 2015-2022, The GAP Group ## gap> ReadPackage( "utils", "tst/loadall.g" );; gap> UtilsLoadingComplete; true ## SubSection 5.1.1 gap> Comm( [ (1,2), (2,3) ] ); (1,2,3) gap> Comm( [(1,2),(2,3),(3,4),(4,5),(5,6)] ); (1,5,6) gap> Comm(Comm(Comm(Comm((1,2),(2,3)),(3,4)),(4,5)),(5,6)); ## the same (1,5,6) ## SubSection 5.1.2 gap> D12 := DihedralGroup( 12 ); gap> SetName( D12, "D12" ); gap> a := D12.1;; b := D12.2;; gap> IsCommuting( a, b ); false ## SubSection 5.1.3 gap> ListOfPowers( 2, 20 ); [ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576 ] gap> ListOfPowers( (1,2,3)(4,5), 12 ); [ (1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), (), (1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), () ] gap> ListOfPowers( D12.2, 6 ); [ f2, f3, f2*f3, f3^2, f2*f3^2, of ... ] ## SubSection 5.1.4 gap> GeneratorsAndInverses( D12 ); [ f1, f2, f3, f1, f2*f3^2, f3^2 ] gap> GeneratorsAndInverses( SymmetricGroup(5) ); [ (1,2,3,4,5), (1,2), (1,5,4,3,2), (1,2) ] ## SubSection 5.1.5 gap> UFS := UpperFittingSeries( D12 );; gap> Set( GeneratorsOfGroup( UFS[2] ) ); [ f3, f2*f3 ] gap> Set( GeneratorsOfGroup( UFS[3] ) ); [ f1, f3, f2*f3 ] gap> LowerFittingSeries( D12 ); [ D12, Group([ f3 ]), Group([ ]) ] gap> FittingLength( D12 ); 2 gap> S4 := SymmetricGroup( 4 );; gap> ufs := UpperFittingSeries( S4 );; gap> List( ufs, StructureDescription ); [ "1", "C2 x C2", "A4", "S4" ] gap> lfs := LowerFittingSeries( S4 );; gap> List( lfs, StructureDescription ); [ "S4", "A4", "C2 x C2", "1" ] gap> FittingLength( S4); 3 ## SubSection 5.2.1 gap> lc1 := LeftCoset( (1,2,3), Group( [ (1,2), (3,4) ] ) ); LeftCoset((1,2,3),Group([ (1,2), (3,4) ])) gap> Representative( lc1 ); (1,2,3) gap> ActingDomain( lc1 ); Group([ (1,2), (3,4) ]) gap> AsSet( lc1 ); [ (2,3), (2,4,3), (1,2,3), (1,2,4,3) ] gap> (1,2,3) in lc1; true gap> lc2 := (2,4,3) * lc1; LeftCoset((1,2,4),Group([ (1,2), (3,4) ])) gap> lc3 := lc1^(2,3,4);; gap> lc2 = lc3; true ## SubSection 5.2.2 gap> rc1 := Inverse( lc1 ); RightCoset(Group([ (1,2), (3,4) ]),(1,3,2)) gap> rc4 := RightCoset( Group( (1,2), (2,3) ), (3,4) ); RightCoset(Group([ (1,2), (2,3) ]),(3,4)) gap> lc4 := Inverse( rc4 ); LeftCoset((3,4),Group([ (1,2), (2,3) ])) gap> Intersection( lc2, lc4 ); [ (2,3,4), (1,2,3,4) ] ## SubSection 5.3.1 gap> G := Group( (1,2,3), (3,4,5), (5,6,7), (7,8,9) );; gap> phi := EpimorphismByGenerators( FreeGroup("a","b","c","d"), G ); [ a, b, c, d ] -> [ (1,2,3), (3,4,5), (5,6,7), (7,8,9) ] gap> PreImagesRepresentativeNC( phi, (1,2,3,4,5,6,7,8,9) ); d*c*b*a gap> a := G.1;; b := G.2;; c := G.3;; d := G.4;; gap> d*c*b*a; (1,2,3,4,5,6,7,8,9) gap> ## note that it is easy to produce nonsense: gap> epi := EpimorphismByGenerators( Group((1,2,3)), Group((8,9)) ); Warning: calling GroupHomomorphismByImagesNC without checks [ (1,2,3) ] -> [ (8,9) ] gap> IsGroupHomomorphism( epi ); true gap> Image(epi,(1,2,3)); () gap> Image(epi,(1,3,2)); (8,9) ## SubSection 5.3.2 gap> s4 := Group( (1,2),(2,3),(3,4) );; gap> s3 := Group( (5,6),(6,7) );; gap> c3 := Subgroup( s3, [ (5,6,7) ] );; gap> f := GroupHomomorphismByImages( s4, s3, > [(1,2),(2,3),(3,4)], [(5,6),(6,7),(5,6)] );; gap> i := GroupHomomorphismByImages( c3, s3, [(5,6,7)], [(5,6,7)] );; gap> Pfi := Pullback( f, i ); Group([ (2,3,4)(5,7,6), (1,2)(3,4) ]) gap> StructureDescription( Pfi ); "A4" gap> info := PullbackInfo( Pfi ); rec( directProduct := Group([ (1,2), (2,3), (3,4), (5,6,7) ]), projections := [ [ (2,3,4)(5,7,6), (1,2)(3,4) ] -> [ (2,3,4), (1,2)(3,4) ], [ (2,3,4)(5,7,6), (1,2)(3,4) ] -> [ (5,7,6), () ] ] ) gap> g := (1,2,3)(5,6,7);; gap> ImageElm( info!.projections[1], g ); (1,2,3) gap> ImageElm( info!.projections[2], g ); (5,6,7) gap> dp := info!.directProduct;; gap> a := ImageElm( Embedding( dp, 1 ), (1,4,3) );; gap> b := ImageElm( Embedding( dp, 2 ), (5,7,6) );; gap> a*b in Pfi; true ## SubSection 5.3.3 gap> g1 := DihedralGroup( 8 ); gap> c1 := Centre( g1 ); Group([ f3 ]) gap> cp1 := CentralProduct( g1, g1, c1, IdentityMapping( c1 ) ); Group([ f1, f2, f5, f3, f4, f5 ]) gap> IdGroup( cp1 ) = IdGroup( ExtraspecialGroup( 2^5, "+" ) ); true gap> g2 := QuaternionGroup( 8 ); gap> c2 := Centre( g2 ); Group([ y2 ]) gap> cp2 := CentralProduct( g2, g2, c2, IdentityMapping( c2 ) ); Group([ f1, f2, f5, f3, f4, f5 ]) gap> IdGroup( cp2 ) = IdGroup( ExtraspecialGroup( 2^5, "+" ) ); true gap> info2 := CentralProductInfo( cp2 ); rec( phi := IdentityMapping( Group([ y2 ]) ), projection := [ f1, f2, f3, f4, f5, f6 ] -> [ f1, f2, f5, f3, f4, f5 ] ) gap> Source( Embedding( Source( info2.projection ), 1 ) ) = g2; true gap> g3 := SymmetricGroup( 3 ); Sym( [ 1 .. 3 ] ) gap> c3 := TrivialSubgroup( g3 ); Group(()) gap> cp3 := CentralProduct( g3, g3, c3, IdentityMapping( c3 ) ); Group([ (1,2,3), (1,2), (4,5,6), (4,5) ]) gap> info3 := CentralProductInfo( cp3 ); rec( phi := IdentityMapping( Group(()) ), projection := IdentityMapping( Group([ (1,2,3), (1,2), (4,5,6), (4,5) ]) ) ) gap> Source( Embedding( Source( info3.projection ), 1 ) ) = g3; true ## SubSection 5.3.4 gap> gens := [ (1,2,3,4), (1,2)(3,4) ];; gap> d8 := Group( gens );; gap> SetName( d8, "d8" ); gap> c2 := Subgroup( d8, [ (2,4) ] );; gap> idemc2 := IdempotentEndomorphismsWithImage( gens, c2 );; gap> idcopy := [ [ (), (2,4) ], [ (2,4), () ] ];; gap> Sort( idemc2 ); gap> Sort( idcopy ); gap> idemc2 = idcopy; true gap> ## depending on packages loaded, the order of images can vary gap> ## so we introduce a convoluted way of checking the images gap> data := IdempotentEndomorphismsData( d8 );; gap> data!.gens; [ (1,2,3,4), (1,2)(3,4) ] gap> images := data!.images;; gap> len := Length( images ); 6 gap> image2 := [ [ [ (), () ] ], [ [ (), (2,4) ], [ (2,4), () ] ], > [ [ (), (1,3) ], [ (1,3), () ] ], > [ [ (), (1,2)(3,4) ], [ (1,2)(3,4), (1,2)(3,4) ] ], > [ [ (), (1,4)(2,3) ], [ (1,4)(2,3), (1,4)(2,3) ] ], > [ [ (1,2,3,4), (1,2)(3,4) ] ] ];; gap> ok := true;; gap> ## new in gapdev - the test 'im in im2' is parsed before the for loop gap> ## so that im2 is an 'unbound global variable' if not declared first gap> im2 := [ ];; gap> for i in [1..len] do > imi := images[i]; > im2 := image2[i]; > if not ForAll( imi, im -> ( im in im2 ) ) then > ok := false; > fi; > od; gap> ok; true gap> List( images, L -> Length(L) ); [ 1, 2, 2, 2, 2, 1 ] gap> idem := IdempotentEndomorphisms( d8 );; gap> idemim := List( idem, m -> MappingGeneratorsImages(m)[2] );; gap> idemim2 := > [ [ (), () ], [ (), (2,4) ], [ (2,4), () ], [ (), (1,3) ], [ (1,3), () ], > [ (), (1,2)(3,4) ], [ (1,2)(3,4), (1,2)(3,4) ], [ (), (1,4)(2,3) ], > [ (1,4)(2,3), (1,4)(2,3) ], [ (1,2,3,4), (1,2)(3,4) ] ];; gap> ForAll( idemim, m -> ( m in idemim2 ) ); true ## SubSection 5.3.5 gap> c4 := Group( (1,2,3,4) );; gap> c2 := Group( (5,6) );; gap> f1 := GroupHomomorphismByImages( c4, c2, [(1,2,3,4)], [(5,6)] );; gap> c3 := Group( (1,2,3) );; gap> c6 := Group( (1,2,3,4,5,6) );; gap> f2 := GroupHomomorphismByImages( c3, c6, [(1,2,3)], [(1,3,5)(2,4,6)] );; gap> c4c3 := DirectProduct( c4, c3 ); Group([ (1,2,3,4), (5,6,7) ]) gap> c2c6 := DirectProduct( c2, c6 ); Group([ (1,2), (3,4,5,6,7,8) ]) gap> f := DirectProductOfFunctions( c4c3, c2c6, f1, f2 ); [ (1,2,3,4), (5,6,7) ] -> [ (1,2), (3,5,7)(4,6,8) ] gap> ImageElm( f, (1,4,3,2)(5,7,6) ); (1,2)(3,7,5)(4,8,6) ## SubSection 5.3.6 gap> c9 := Group( (1,2,3,4,5,6,7,8,9) );; gap> ac9 := AutomorphismGroup( c9 );; gap> q8 := QuaternionGroup( IsPermGroup, 8 );; gap> aq8 := AutomorphismGroup( q8 );; gap> A := DirectProductOfAutomorphismGroups( ac9, aq8 ); gap> genA := GeneratorsOfGroup( A );; gap> G := Source( genA[1] ); Group([ (1,2,3,4,5,6,7,8,9), (10,14,12,16)(11,17,13,15), (10,11,12,13) (14,15,16,17) ]) gap> a := genA[1]*genA[5]; [ (1,2,3,4,5,6,7,8,9), (10,14,12,16)(11,17,13,15), (10,11,12,13)(14,15,16,17) ] -> [ (1,3,5,7,9,2,4,6,8), (10,16,12,14)(11,15,13,17), (10,11,12,13)(14,15,16,17) ] gap> ImageElm( a, (1,9,8,7,6,5,4,3,2)(10,14,12,16)(11,17,13,15) ); (1,8,6,4,2,9,7,5,3)(10,16,12,14)(11,15,13,17) utils-0.86/tst/loadall.g000755 000765 000024 00000001236 14621444241 020314 0ustar00christopherwensleystaff000000 000000 ############################################################################## ## #W loadall.g Utils Package Chris Wensley ## #Y Copyright (C) 2015-2016, The GAP Group ## ## first make sure that, if the transferred code has not been read, ## then the appropriate packages are loaded. if not UtilsLoadingComplete then len := Length( UtilsPackageVersions ); j := 0; while ( j < len ) do j := j+1; name := UtilsPackageVersions[j]; if not OKtoReadFromUtils( name ) then LoadPackage( name ); fi; j := j+1; od; UtilsLoadingComplete := true; fi; utils-0.86/tst/download.tst000644 000765 000024 00000006232 14621444241 021075 0ustar00christopherwensleystaff000000 000000 #@local meths, i, urls, pair, url, expected, res1, good1, n, file, res2, good2, contents, r; ############################################################################ ## #W download.tst Utils Package Thomas Breuer ## #Y Copyright (C) 2022, The GAP Group ## gap> ReadPackage( "utils", "tst/loadall.g" );; gap> UtilsLoadingComplete; true ## Test the available Download methods gap> meths:= List( Filtered( Download_Methods, r -> r.isAvailable() ), > ShallowCopy );; gap> for i in [ 1 .. Length( meths ) ] do > meths[i].position:= String( i ); > od; gap> urls:= [ # a http url that gets redirected to https > [ "http://www.gap-system.org/Packages/utils.html", true ], > # a http url that works as such > [ "http://www.math.rwth-aachen.de/index.html", true ], > # a https url that exists > [ "https://www.gap-system.org/Packages/utils.html", true ], > # a https url that does not exist > [ "https://www.gap-system.org/Packages/utilsxxx.html", false ], > ];; ## The problem is that the methods do not behave consistently ## in the case of failure. ## (Well, they even do not agree what failure means.) ## The test results depend on which methods are available at runtime, ## which makes them useless as automatic tests. ## Thus we test only working http and https urls. gap> urls:= urls{ [ 2, 3 ] };; gap> for pair in urls do > url:= pair[1]; > expected:= pair[2]; > res1:= List( meths, r -> [ r.download( url, rec() ), r.position ] );; > good1:= Filtered( res1, r -> r[1].success = true );; > if expected = false and Length( good1 ) > 0 then > Print( "success for url ", url, "?\n" ); > fi; > n:= Length( Set( good1, r -> r[1].result ) ); > if n > 1 then > Print( "different results (", n, ") for url ", url, "\n" ); > fi; > file:= Filename( DirectoryTemporary(), "test" ); > res2:= List( meths, > r -> [ r.download( url, > rec( target:= Concatenation( file, r.position ) ) ), > r.position ] );; > good2:= Filtered( res2, r -> r[1].success = true );; > if expected = false and Length( good2 ) > 0 then > Print( "success for url ", url, "?\n" ); > fi; > if Length( good1 ) <> Length( good2 ) then > Print( "different success cases for url ", url, "\n" ); > fi; > if Length( good1 ) > 0 then > contents:= good1[1][1].result; > for r in good2 do > if contents <> StringFile( Concatenation( file, r[2] ) ) then > Print( "different files and contents for url ", url, "\n" ); > fi; > od; > fi; > od; ## the example 9.1.1 from the manual gap> url:= "https://www.gap-system.org/Packages/utils.html";; gap> res1:= Download( url );; gap> res1.success; true gap> IsBound( res1.result ) and IsString( res1.result ); true gap> res2:= Download( Concatenation( url, "xxx" ) );; gap> res2.success; false gap> IsBound( res2.error ) and IsString( res2.error ); true ############################################################################# ## #E utils-0.86/tst/number.tst000644 000765 000024 00000005423 14621444241 020557 0ustar00christopherwensleystaff000000 000000 #@local i, iter, n, p, sum ############################################################################## ## #W number.tst Utils Package Stefan Kohl ## #Y Copyright (C) 2015-2022, The GAP Group ## gap> ReadPackage( "utils", "tst/loadall.g" );; gap> UtilsLoadingComplete; true ## SubSection 4.1.1 gap> AllSmoothIntegers( 3, 1000 ); [ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576, 648, 729, 768, 864, 972 ] gap> AllSmoothIntegers( [5,11,17], 1000 ); [ 1, 5, 11, 17, 25, 55, 85, 121, 125, 187, 275, 289, 425, 605, 625, 935 ] gap> Length( last ); 16 gap> List( [3..20], n -> Length( AllSmoothIntegers( [5,11,17], 10^n ) ) ); [ 16, 29, 50, 78, 114, 155, 212, 282, 359, 452, 565, 691, 831, 992, 1173, 1374, 1595, 1843 ] ## SubSection 4.1.2 gap> AllProducts([1..4],3); [ 1, 2, 3, 4, 2, 4, 6, 8, 3, 6, 9, 12, 4, 8, 12, 16, 2, 4, 6, 8, 4, 8, 12, 16, 6, 12, 18, 24, 8, 16, 24, 32, 3, 6, 9, 12, 6, 12, 18, 24, 9, 18, 27, 36, 12, 24, 36, 48, 4, 8, 12, 16, 8, 16, 24, 32, 12, 24, 36, 48, 16, 32, 48, 64 ] gap> Set(last); [ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 64 ] gap> AllProducts( [(1,2,3),(2,3,4)], 2 ); [ (2,4,3), (1,2)(3,4), (1,3)(2,4), (1,3,2) ] ## SubSection 4.1.3 gap> RestrictedPartitions( 20, [4..10] ); [ [ 4, 4, 4, 4, 4 ], [ 5, 5, 5, 5 ], [ 6, 5, 5, 4 ], [ 6, 6, 4, 4 ], [ 7, 5, 4, 4 ], [ 7, 7, 6 ], [ 8, 4, 4, 4 ], [ 8, 6, 6 ], [ 8, 7, 5 ], [ 8, 8, 4 ], [ 9, 6, 5 ], [ 9, 7, 4 ], [ 10, 5, 5 ], [ 10, 6, 4 ], [ 10, 10 ] ] gap> RestrictedPartitionsWithoutRepetitions( 20, [4..10] ); [ [ 10, 6, 4 ], [ 9, 7, 4 ], [ 9, 6, 5 ], [ 8, 7, 5 ] ] gap> RestrictedPartitionsWithoutRepetitions( 10^2, List([1..10], n->n^2 ) ); [ [ 100 ], [ 64, 36 ], [ 49, 25, 16, 9, 1 ] ] ## SubSection 4.1.4 gap> n := 2^251;; gap> NextProbablyPrimeInt( n ); 3618502788666131106986593281521497120414687020801267626233049500247285301313 ## SubSection 4.1.6 gap> iter := PrimeNumbersIterator();; gap> for i in [1..100] do p := NextIterator(iter); od; gap> p; 541 gap> sum := 0;; gap> ## "prime number race" 1 vs. 3 mod 4 gap> for p in PrimeNumbersIterator() do > if p <> 2 then sum := sum + E(4)^(p-1); fi; > if sum > 0 then break; fi; > od; gap> p; 26861 ## this final example takes quite a while: use examples/number.g ## gap> sum := 0;; ## gap> ## "prime number race" 1 vs. 5 mod 8 ## gap> for p in PrimeNumbersIterator() do ## > if p mod 8 in [1,5] then sum := sum + E(4)^((p-1)/2); fi; ## > if sum > 0 then break; fi; ## > od; ## gap> p; ## 588067889 ############################################################################# ## #E number.tst . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/tst/record.tst000644 000765 000024 00000004541 14621444241 020545 0ustar00christopherwensleystaff000000 000000 #@local r, defaults, PrintDimensions, mydim ############################################################################## ## #W record.tst Utils Package ## #Y Copyright (C) 2015-2022, The GAP Group ## gap> ReadPackage( "utils", "tst/loadall.g" );; gap> UtilsLoadingComplete; true ## SubSection 8.1.1 gap> r := rec( a := 1, b := 2, c := 3 );; gap> AssignGlobals( r ); The following global variables have been assigned: [ "a", "b", "c" ] gap> [ a, b, c ]; [ 1, 2, 3 ] ## SubSection 8.2.1 gap> defaults := rec( a := 1, b := 2, c := 3 );; gap> OptionRecordWithDefaults( defaults, rec( a := 6) ); rec( a := 6, b := 2, c := 3 ) gap> OptionRecordWithDefaults( defaults, rec( b := 7, c := 8 ) ); rec( a := 1, b := 7, c := 8 ) gap> OptionRecordWithDefaults( defaults, [ ] ); rec( a := 1, b := 2, c := 3 ) gap> OptionRecordWithDefaults( defaults, [ rec( c := 8 ) ] ); rec( a := 1, b := 2, c := 8 ) gap> OptionRecordWithDefaults( defaults, rec( d := 9 ) ); Error, Unknown option: d gap> OptionRecordWithDefaults( defaults, [ rec( b := 7 ), rec( c := 8 ) ] ); Error, Too many arguments for function gap> OptionRecordWithDefaults( defaults, 5 ); Error, Options should be a record gap> OptionRecordWithDefaults( defaults, [6,7,8] ); Error, Too many arguments for function gap> PrintDimensions := function( arg ) > local nargs, dim, order, V, L, len, K, i; > nargs := Length( arg ); > dim := [ arg[1]!.height, arg[1]!.width, arg[1]!.depth ]; > order := rec( h := 1, w := 2, d := 3 ); > V := [ "height", "width", "depth" ]; > if ( nargs > 1 ) and IsRecord( arg[2] ) then > order := OptionRecordWithDefaults( order, arg[2] ); > fi; > L := [ order!.h, order!.w, order!.d ]; > len := Length( L ); > K := [ 1..len ]; > SortParallel( L, K ); > Print( "dimensions: " ); > Print( V[K[1]], " = ", dim[K[1]], ", " ); > Print( V[K[2]], " = ", dim[K[2]], ", " ); > Print( V[K[3]], " = ", dim[K[3]], "\n" ); > end;; gap> mydim := rec( height := 45, width := 31, depth := 17 ); rec( depth := 17, height := 45, width := 31 ) gap> PrintDimensions( mydim ); dimensions: height = 45, width = 31, depth = 17 gap> PrintDimensions( mydim, rec( h:=3, w:=1, d:=2 ) ); dimensions: width = 31, depth = 17, height = 45 utils-0.86/tst/matrix.tst000644 000765 000024 00000005130 14621444241 020566 0ustar00christopherwensleystaff000000 000000 #@local M6, L6, M4, L4, M8, L8, L, A, M3, L3, M5, L5; ############################################################################## ## #W matrix.tst Utils Package ## #Y Copyright (C) 2015-2023, The GAP Group ## gap> ReadPackage( "utils", "tst/loadall.g" );; gap> UtilsLoadingComplete; true ## SubSection 9.1.1 gap> M6 := [ [1,2,0,0,0,0], [3,4,0,0,0,0], [5,6,0,0,0,0], > [0,0,9,0,0,0], [0,0,0,1,2,3], [0,0,0,4,5,6] ];; gap> Display( M6 ); [ [ 1, 2, 0, 0, 0, 0 ], [ 3, 4, 0, 0, 0, 0 ], [ 5, 6, 0, 0, 0, 0 ], [ 0, 0, 9, 0, 0, 0 ], [ 0, 0, 0, 1, 2, 3 ], [ 0, 0, 0, 4, 5, 6 ] ] gap> L6 := DirectSumDecompositionMatrices( M6 ); [ [ [ [ 1, 2 ], [ 3, 4 ], [ 5, 6 ] ], [ [ 9 ] ], [ [ 1, 2, 3 ], [ 4, 5, 6 ] ] ] ] gap> M4 := [ [0,3,0,0], [0,0,0,0], [0,0,0,0], [0,0,4,0] ];; gap> Display( M4 ); [ [ 0, 3, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 4, 0 ] ] gap> L4 := DirectSumDecompositionMatrices( M4 ); [ [ [ [ 0, 3 ] ], [ [ 0, 0 ], [ 0, 0 ], [ 4, 0 ] ] ], [ [ [ 0, 3 ], [ 0, 0 ] ], [ [ 0, 0 ], [ 4, 0 ] ] ], [ [ [ 0, 3 ], [ 0, 0 ], [ 0, 0 ] ], [ [ 4, 0 ] ] ] ] gap> for L in L4 do > A := DirectSumMat( L );; > if ( A = M4 ) then Print( "yes, A = M4\n" ); fi; > od; yes, A = M4 yes, A = M4 yes, A = M4 gap> M8 := DirectSumMat( M4, M4 );; gap> Display( M8 ); [ [ 0, 3, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 4, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 3, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 4, 0 ] ] gap> L8 := DirectSumDecompositionMatrices( M8 );; gap> Length( L8 ); 16 gap> M3 := [ [0,0,0,7,0,0,0], [0,0,0,0,8,0,0], [0,0,0,0,0,9,0] ];; gap> Display( M3 ); [ [ 0, 0, 0, 7, 0, 0, 0 ], [ 0, 0, 0, 0, 8, 0, 0 ], [ 0, 0, 0, 0, 0, 9, 0 ] ] gap> L3 := DirectSumDecompositionMatrices( M3 ); [ [ [ [ 0, 0, 0, 7 ] ], [ [ 8 ] ], [ [ 9, 0 ] ] ] ] gap> M5 := [ [1,2,0,0,0], [3,4,0,0,0], [0,0,0,0,0], > [0,0,0,6,7], [0,0,0,8,9] ];; gap> Display(M5); [ [ 1, 2, 0, 0, 0 ], [ 3, 4, 0, 0, 0 ], [ 0, 0, 0, 0, 0 ], [ 0, 0, 0, 6, 7 ], [ 0, 0, 0, 8, 9 ] ] gap> L5 := DirectSumDecompositionMatrices( M5 ); [ [ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 0 ] ], [ [ 6, 7 ], [ 8, 9 ] ] ] ] ############################################################################# ## #E matrix.tst . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/tst/testall.g000755 000765 000024 00000001145 14621444241 020353 0ustar00christopherwensleystaff000000 000000 ############################################################################ ## #W testall.g Utils Package Chris Wensley ## #Y Copyright (C) 2015-2023, The GAP Group ## LoadPackage( "utils" ); ## first make sure that, if the transferred code has not been read, ## then the appropriate packages are loaded. if not UtilsLoadingComplete then ReadPackage( "utils", "tst/loadall.g" ); fi; dir := DirectoriesPackageLibrary( "utils", "tst" ); TestDirectory(dir, rec(exitGAP := true, testOptions:=rec(compareFunction := "uptowhitespace"))); FORCE_QUIT_GAP(1); utils-0.86/tst/others.tst000644 000765 000024 00000003546 14621444241 020577 0ustar00christopherwensleystaff000000 000000 #@local M, s1, n1, n2, N, s2; ############################################################################ ## #W others.tst Utils Package ## #Y Copyright (C) 2015-2022, The GAP Group ## gap> ReadPackage( "utils", "tst/loadall.g" );; gap> UtilsLoadingComplete; true ## SubSection 10.1.1 ## this manual example is not tested to avoid creating files triv.* ## gap> LogTo( "triv.log" ); ## gap> a := 33^5; ## 39135393 ## gap> LogTo(); ## gap> Log2HTML( "triv.log" ); ## SubSection 10.2.1 gap> IntOrInfinityToLaTeX( 10^3 ); "1000" gap> IntOrInfinityToLaTeX( infinity ); "\\infty" ## SubSection 10.2.2 gap> LaTeXStringFactorsInt( Factorial(12) ); "2^{10} \\cdot 3^5 \\cdot 5^2 \\cdot 7 \\cdot 11" ## SubSection 10.3.1 gap> ConvertToMagmaInputString( Group( (1,2,3,4,5), (3,4,5) ) ); "PermutationGroup<5|(1,2,3,4,5),\n(3,4,5)>;\n" gap> ConvertToMagmaInputString( Group( (1,2,3,4,5) ), "c5" ); "c5 := PermutationGroup<5|(1,2,3,4,5)>;\n" gap> ConvertToMagmaInputString( DihedralGroup( IsPcGroup, 10 ) ); "PolycyclicGroup< f1,f2 |\nf1^2,\nf2^5,\nf2^f1 = f2^4\n>;\n" gap> M := GL(2,5);; Size(M); 480 gap> s1 := ConvertToMagmaInputString( M ); "F := GF(5);\nP := GL(2,F);\ngens := [\nP![2,0,0,1],\nP![4,1,4,0]\n];\nsub

;\n" gap> Print( s1 ); F := GF(5); P := GL(2,F); gens := [ P![2,0,0,1], P![4,1,4,0] ]; sub

; gap> n1 := [ [ Z(9)^0, Z(9)^0 ], [ Z(9)^0, Z(9) ] ];; gap> n2 := [ [ Z(9)^0, Z(9)^3 ], [ Z(9)^4, Z(9)^2 ] ];; gap> N := Group( n1, n2 );; Size( N ); 5760 gap> s2 := ConvertToMagmaInputString( N, "gpN" );; gap> Print( s2 ); F := GF(3^2); P := GL(2,F); w := PrimitiveElement(F); gens := [ P![ 1, 1, 1,w^1], P![ 1,w^3, 2,w^2] ]; gpN := sub

; ############################################################################# ## #E others.tst . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/tst/lcset.tst000644 000765 000024 00000012266 14621444241 020404 0ustar00christopherwensleystaff000000 000000 #@local a, b, d, f, g, h, H1, H2, H3, K1, K2, K3, lc1, lc4, M, matcyc, rc5, s ############################################################################# ## adapted from gapdev/tst/tstinstall/cset.tst for left cosets ## ## test of group intersection and LeftCoset ## gap> START_TEST("lcset.tst"); # basic coset tests gap> LeftCoset( (1,2), Group([(1,2,3),(2,3,4)]) ) > = LeftCoset( (2,3), Group([(1,2,3),(2,3,4)]) ); true gap> () in LeftCoset( (1,2), Group([(1,2,3,4)]) ); false gap> (1,2) in LeftCoset( (5,6), SymmetricGroup(12) ); true gap> Length( LeftCosets( SymmetricGroup(5), AlternatingGroup(4) ) ); 10 gap> (1,2,3) * LeftCoset( (), AlternatingGroup(4) ) > = LeftCoset( (), AlternatingGroup(4) ); true gap> IsBiCoset( LeftCoset( (1,2), AlternatingGroup(6) ) ); true gap> IsBiCoset( LeftCoset( (1,7), AlternatingGroup(6) ) ); false gap> IsLeftCoset( LeftCoset( (1,2,3), MathieuGroup(12) ) ); true # test intersecting permutation cosets gap> H1 := Group( [ (), (2,7,6)(3,4,5), (1,2,7,5,6,4,3) ] );; gap> H2 := Group( [ (1,2,3,4,5,6,7), (5,6,7) ] );; gap> H3 := Group( [ (1,2,3,4,5,6,8), (1,3,2,6,4,5), (1,6)(2,3)(4,5)(7,8) ] );; gap> AsSet( LeftCoset( (1,5,7,3)(4,6), H1 ) ) = > Intersection( LeftCoset( (3,6)(4,7), H2 ), > LeftCoset( (1,5,3,8,6,7), H3 ) ); true gap> AsSet( LeftCoset( (1,2,5,6,7,4,3,8), Group(()) ) ) = > Intersection( LeftCoset( (1,5,6,7)(3,8,4), > Group( [ (1,4)(2,5), (1,3,5)(2,4,6), (1,5)(2,4)(3,6) ] ) ), > LeftCoset( (1,2,6,8)(3,7), > Group( [ (3,4), (5,6,7,8), (5,6) ] ) ) ); true gap> [] = Intersection( LeftCoset( (), SymmetricGroup(4) ), > LeftCoset( (4,7), SymmetricGroup([3..6]) ) ); true gap> [] = Intersection( LeftCoset( (4,5), Group( [ (1,2,3,4,5) ] ) ), > LeftCoset( (), AlternatingGroup(4) ) ); true gap> AsSet( LeftCoset( (7,9), SymmetricGroup([3..5]) ) ) = > Intersection( LeftCoset( (1,2)(7,9), SymmetricGroup(5) ), > LeftCoset( (7,9), SymmetricGroup([3..7]) ) ); true gap> [] = Intersection( LeftCoset( (1,4)(3,5), Group([(1,2,3,4,5)]) ), > LeftCoset( (), SymmetricGroup(3) ) ); true gap> AsSet( LeftCoset( (4,5), Group( [(5,6)] ) ) ) = > Intersection( LeftCoset( (), SymmetricGroup(6) ), > LeftCoset( (4,5), SymmetricGroup([5..8]) ) ); true gap> AsSet( LeftCoset( (1,4,5), SymmetricGroup(5) ) ) = > Intersection( LeftCoset( (), SymmetricGroup(5) ), > LeftCoset( (1,2), SymmetricGroup(5) ) ); true gap> [] = > Intersection( LeftCoset( (1,2), Group( (1,2,3,4,5) ) ), > LeftCoset( (), Group( (1,2,3,5,4) ) ) ); true gap> [] = > Intersection( LeftCoset( (1,2,3), Group( (1,2,3,4,5) ) ), > LeftCoset( (), Group( (1,2,3,5,4) ) ) ); true gap> AsSet( LeftCoset( (1,2), Group( [ (1,2,3,5,4) ] ) ) ) = > Intersection( LeftCoset( (), SymmetricGroup(7) ), > LeftCoset( (1,2), Group( (1,2,3,5,4) ) ) ); true gap> [] = > Intersection( LeftCoset( (), SymmetricGroup([3..7]) ), > LeftCoset( (1,2), Group( (1,2,3,5,4) ) ) ); true # test trivial cases gap> Intersection( LeftCoset( (), Group([],()) ), > LeftCoset( (1,2), Group([],()) ) ) = []; true gap> Intersection( LeftCoset( (), Group( (1,2,3) ) ), > LeftCoset( (1,2), Group( (1,2,3) ) ) ) = []; true gap> Intersection( LeftCoset( (), AlternatingGroup(6) ), > LeftCoset( (1,2), AlternatingGroup(6) ) ) = []; true gap> Intersection( LeftCoset( (1,2), AlternatingGroup([1..5]) ), > LeftCoset( (1,2), AlternatingGroup([6..10]) ) ) > = AsSet( LeftCoset( (1,2), Group(()) ) ); true #coset of pc-group gap> d := DihedralGroup( 24 ); gap> List( GeneratorsOfGroup(d), x -> Order(x) ); [ 2, 12, 6, 3 ] gap> s := Subgroup( d, [ d.1, d.4 ] );; gap> SetName( s, "s" ); gap> lc4 := LeftCoset( d.2, s ); LeftCoset(,s) gap> AsSet( lc4 ); [ f2, f2*f4, f1*f2*f3, f2*f4^2, f1*f2*f3*f4, f1*f2*f3*f4^2 ] gap> d.2 * d.4 in lc4; true # coset of fp-group gap> f := FreeGroup(2);; a := f.1;; b := f.2;; gap> g := f / [ a^5, b^4, a*b*a^2*b^3 ]; gap> Size(g); 20 gap> h := Subgroup( g, [g.1] );; gap> SetName( h, "C5" ); gap> rc5 := LeftCoset( g.2, h ); LeftCoset(,C5) gap> AsSet( rc5 ); [ f2, f2*f1, f2*f1^2, f2*f1^3, f2*f1^4 ] # test intersection non-permutation cosets gap> K1 := Group( [ [[-1,0],[0,-1]] ] );; gap> K2 := Group( [ [[-1,0],[0,1]], [[0,1],[1,0]] ] );; gap> K3 := Group( [ - IdentityMat(2) ] );; gap> AsSet( LeftCoset( [ [0,1],[1,0] ], K1 ) ) = > Intersection( LeftCoset( IdentityMat(2), K2 ), > LeftCoset( [[0,1],[1,0]], K1 ) ); true gap> AsSet( LeftCoset( [[0,1],[1,0]], K3 ) ) = > Intersection( LeftCoset( [[0,-1],[-1,0]], K3 ), > LeftCoset( [[0,1],[1,0]], K3 ) ); true gap> matcyc := CyclicGroup( IsMatrixGroup, GF(3), 4 );; gap> M := GeneratorsOfGroup( matcyc )[1];; gap> lc1 := LeftCoset( M^2, matcyc );; gap> Representative(lc1); [ [ 0*Z(3), 0*Z(3), Z(3)^0, 0*Z(3) ], [ 0*Z(3), 0*Z(3), 0*Z(3), Z(3)^0 ], [ Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3), 0*Z(3) ] ] # gap> STOP_TEST("lcset.tst", 1); utils-0.86/doc/chap9_mj.html000644 000765 000024 00000022604 14621610515 021045 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Chapter 9: Web Downloads
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9 Web Downloads
 9.1 Functions for downloading files from the web

  9.1-1 Download
  9.1-2 User preference DownloadVerifyCertificate

9 Web Downloads

The Download operation has been written by Thomas Breuer, incorporating a number of suggestions from Max Horn, for version 0.77 of Utils. It implements downloading a file from within GAP. It can use the IO or curlInterface packages, or wget or curl, if installed, and it can be extended with other download methods quite easily. It is envisaged that, once other packages have started to use it, and any problems have been addressed, that the functions will be transferred to the main GAP library.

9.1 Functions for downloading files from the web

9.1-1 Download
‣ Download( url[, opt] )( function )

This function downloads the file with the web address url, which must be a string.

The result is a record which has at least the component success, with value true if the download was successful and false otherwise. In the former case, the component result is bound, whose value is a string that contains the contents of the downloaded file. In the latter case, the component error is bound, whose value is a string that describes the problem.

The function calls the methods stored in the global list Download_Methods until one of them is successful. Currently there are methods based on the GAP functions DownloadURL (curl: DownloadURL) and SingleHTTPRequest (IO: SingleHTTPRequest), and methods based on the external programs wget and curl.

An optional record opt can be given. The following components are supported.

target

If this component is bound then its value must be a string that is a local filename, and the function writes the downloaded contents to this file; the returned record does not have a result component in this case.

verifyCert

If this component is bound and has the value false then those download methods that are based on curl or wget will omit the check of the server's certificate. The same effect is achieved for all Download calls by setting the user preference DownloadVerifyCertificate (see 9.1-2) to false and omitting the verifyCert component from opt.


gap> url:= "https://www.gap-system.org/Packages/utils.html";;
gap> res1:= Download( url );;
gap> res1.success;
true
gap> IsBound( res1.result ) and IsString( res1.result );
true
gap> res2:= Download( Concatenation( url, "xxx" ) );;
gap> res2.success;
false
gap> IsBound( res2.error ) and IsString( res2.error );
true

9.1-2 User preference DownloadVerifyCertificate

The value true (the default) means that the server's certificate is checked in calls of Download (9.1-1), such that nothing gets downloaded if the certificate is invalid.

If the value is false then download methods are supposed to omit the check of the server's certificate (this may not be supported by all download methods).

One can set the value of the preference to be val via SetUserPreference (Reference: SetUserPreference), by calling SetUserPreference( "utils", "DownloadVerifyCertificate", val ), and access the current value via UserPreference (Reference: UserPreference), by calling UserPreference( "utils", "DownloadVerifyCertificate" ).

We recommend leaving this preference at its default value true. Sometimes it can be necessary to change it, e.g. to work around issues with old operating systems which may not be able to correctly verify new certificates. In general it is better to update such a system, but if that is not an option, then disabling certificate checks may be a good last resort.

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"\033[2XPullback\033[102X", "5.3-2", [ 5, 3, 2 ], 221, 20, "pullback", "X7C705F2A79F8E43C" ], [ "\033[2XPullbackInfo\033[102X", "5.3-2", [ 5, 3, 2 ], 221, 20, "pullbackinfo", "X7C705F2A79F8E43C" ], [ "\033[2XCentralProduct\033[102X", "5.3-3", [ 5, 3, 3 ], 272, 20, "centralproduct", "X78DD2C617B992BE2" ], [ "\033[2XCentralProductInfo\033[102X", "5.3-3", [ 5, 3, 3 ], 272, 20, "centralproductinfo", "X78DD2C617B992BE2" ], [ "\033[2XIdempotentEndomorphisms\033[102X", "5.3-4", [ 5, 3, 4 ], 323, 21, "idempotentendomorphisms", "X801038CB808FC956" ], [ "\033[2XIdempotentEndomorphismsData\033[102X", "5.3-4", [ 5, 3, 4 ], 323, 21, "idempotentendomorphismsdata", "X801038CB808FC956" ], [ "\033[2XIdempotentEndomorphismsWithImage\033[102X", "5.3-4", [ 5, 3, 4 ], 323, 21, "idempotentendomorphismswithimage", "X801038CB808FC956" ], [ "\033[2XDirectProductOfFunctions\033[102X", "5.3-5", [ 5, 3, 5 ], 391, 22, "directproductoffunctions", "X81FA9E6C7F3B9238" ], [ "\033[2XDirectProductOfAutomorphismGroups\033[102X", "5.3-6", [ 5, 3, 6 ], 418, 23, "directproductofautomorphismgroups", "X7CB2D5F27F4182AF" ], [ "\033[2XDirectSumDecompositionMatrices\033[102X", "6.1-1", [ 6, 1, 1 ], 7, 24, "directsumdecompositionmatrices", "X787B89237E1398B6" ], [ "Iterators", "7.1", [ 7, 1, 0 ], 4, 26, "iterators", "X7BB5350081B27D17" ] , [ "\033[2XAllIsomorphismsIterator\033[102X", "7.1-1", [ 7, 1, 1 ], 11, 26, "allisomorphismsiterator", "X7F8B54D1806C762D" ], [ "\033[2XAllIsomorphismsNumber\033[102X", "7.1-1", [ 7, 1, 1 ], 11, 26, "allisomorphismsnumber", "X7F8B54D1806C762D" ], [ "\033[2XAllIsomorphisms\033[102X", "7.1-1", [ 7, 1, 1 ], 11, 26, "allisomorphisms", "X7F8B54D1806C762D" ], [ "\033[2XAllSubgroupsIterator\033[102X", "7.1-2", [ 7, 1, 2 ], 46, 27, "allsubgroupsiterator", "X831DA5AE8437578F" ], [ "\033[2XCartesianIterator\033[102X", "7.2-1", [ 7, 2, 1 ], 80, 27, "cartesianiterator", "X87395A9181A35301" ], [ "\033[2XUnorderedPairsIterator\033[102X", "7.2-2", [ 7, 2, 2 ], 105, 28, "unorderedpairsiterator", "X7C95E27987A812EA" ], [ "\033[2XAssignGlobals\033[102X", "8.1-1", [ 8, 1, 1 ], 7, 29, "assignglobals", "X84D82EB579B2ACCD" ], [ "\033[2XOptionRecordWithDefaults\033[102X", "8.2-1", [ 8, 2, 1 ], 30, 29, "optionrecordwithdefaults", "X8322B9377CC590D2" ], [ "\033[2XDownload\033[102X", "9.1-1", [ 9, 1, 1 ], 15, 32, "download", "X7A7438AE8448635E" ], [ "\033[10XDownloadVerifyCertificate\033[110X", "9.1-2", [ 9, 1, 2 ], 65, 33, "downloadverifycertificate", "X85182BA486E3C2AA" ], [ "\033[2XLog2HTML\033[102X", "10.1-1", [ 10, 1, 1 ], 7, 34, "log2html", "X7B7ECADF85F748BE" ], [ "\033[2XIntOrOnfinityToLaTeX\033[102X", "10.2-1", [ 10, 2, 1 ], 32, 34, "intoronfinitytolatex", "X87DEB2B58266F858" ], [ "\033[2XLaTeXStringFactorsInt\033[102X", "10.2-2", [ 10, 2, 2 ], 49, 35, "latexstringfactorsint", "X7DC642B97CD02F4E" ], [ "\033[2XConvertToMagmaInputString\033[102X", "10.3-1", [ 10, 3, 1 ], 68, 35, "converttomagmainputstring", "X8768D7707B4CBBD4" ], [ "PermGroupToMagmaFormat", "10.3-1", [ 10, 3, 1 ], 68, 35, "permgrouptomagmaformat", "X8768D7707B4CBBD4" ], [ "PcGroupToMagmaFormat", "10.3-1", [ 10, 3, 1 ], 68, 35, "pcgrouptomagmaformat", "X8768D7707B4CBBD4" ], [ "MatrixGroupToMagmaFormat", "10.3-1", [ 10, 3, 1 ], 68, 35, "matrixgrouptomagmaformat", "X8768D7707B4CBBD4" ], [ "FindMatchingFiles", "11.1", [ 11, 1, 0 ], 4, 37, "findmatchingfiles", "X7A6BB3D084912F35" ], [ "CreateDirIfMissing", "11.1", [ 11, 1, 0 ], 4, 37, "createdirifmissing", "X7A6BB3D084912F35" ], [ "StringDotSuffix", "11.1", [ 11, 1, 0 ], 4, 37, "stringdotsuffix", "X7A6BB3D084912F35" ], [ "GetSuffix", "11.1", [ 11, 1, 0 ], 4, 37, "getsuffix", "X7A6BB3D084912F35" ], [ "SetIfMissing", "11.1", [ 11, 1, 0 ], 4, 37, "setifmissing", "X7A6BB3D084912F35" ], [ "PrintOneItemPerLine", "11.2", [ 11, 2, 0 ], 21, 37, "printoneitemperline", "X86F322FC7DECE36F" ], [ "PrintApplicableMethod", "11.3-1", [ 11, 3, 1 ], 52, 38, "printapplicablemethod", "X78B7D1A982BE9866" ], [ "ExponentOfPrime", "11.3-2", [ 11, 3, 2 ], 85, 38, "exponentofprime", "X7C1AF2467FB55D79" ], [ "OKtoReadFromUtils", "12.0", [ 12, 0, 0 ], 1, 39, "oktoreadfromutils", "X84AC9613842F014C" ] ] ); utils-0.86/doc/chapBib_mj.html000644 000765 000024 00000014121 14621610515 021364 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - References
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References

[BCP97] Bosma, W., Cannon, J. and Playoust, C., The Magma algebra system. {I}. The user language, J. Symbolic Comput., 24 (3-4) (1997), 235--265 pages
( Computational algebra and number theory (London, 1993)} https://doi.org/10.1006/jsco.1996.0125 ).

[GH16] Gutsche, S. and Horn, M., AutoDoc - Generate documentation from GAP source code (Version 2016.12.04) (2016)
( GAP package, https://github.com/gap-packages/AutoDoc ).

[Hor17] Horn, M., GitHubPagesForGAP - Template for easily using GitHub Pages within GAP packages (Version 0.2) (2017)
( GAP package, https://gap-system.github.io/GitHubPagesForGAP/ ).

[Koh17a] Kohl, S., RCWA - Residue-Class-Wise Affine Groups (Version 4.5.1) (2017)
( GAP package, https://stefan-kohl.github.io/rcwa.html ).

[Koh17b] Kohl, S., ResClasses - Set-Theoretic Computations with Residue Classes (Version 4.6.0) (2017)
( GAP package, https://stefan-kohl.github.io/resclasses.html ).

[LN17] Lübeck, F. and Neunhöffer, M., GAPDoc (Version 1.6), RWTH Aachen (2017)
( GAP package, https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/index.html ).

[WAOU17] Wensley, C. D., Alp, M., Odabas, A. and Uslu, E. O., XMod - Crossed Modules and Cat1-groups in GAP (Version 2.64) (2017)
( GAP package, https://github.com/gap-packages/xmod ).

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utils-0.86/doc/record.xml000644 000765 000024 00000011615 14621444241 020466 0ustar00christopherwensleystaff000000 000000 Records
Functions for records This function has been transferred from package &RCWA;.

It assigns the record components of rec to global variables with the same names.

r := rec( a := 1, b := 2, c := 3 );; gap> AssignGlobals( r ); The following global variables have been assigned: [ "a", "b", "c" ] gap> [a,b,c]; [ 1, 2, 3 ] ]]>

Option records for functions This functions has been transferred by Chris Jefferson from other packages. It simplifies the handling of records which are intended to be used for expressing configuration options. defaults represents the "default record", and useroptions lets the user give new values for values in defaults.

The function returns a record with the same component names as defaults and which has the same values as defaults, except for those component names in useroptions, where the values in useroptions are used instead. An error is given if useroptions contains any component names not in defaults. If useroptions is an empty list it is treated as an empty record, and if useroptions is a list of length 1 containing a record, this record is used as useroptions.

defaults := rec( a := 1, b := 2, c := 3 );; gap> OptionRecordWithDefaults( defaults, rec( a := 6) ); rec( a := 6, b := 2, c := 3 ) gap> OptionRecordWithDefaults( defaults, rec( b := 7, c := 8 ) ); rec( a := 1, b := 7, c := 8 ) gap> OptionRecordWithDefaults( defaults, [ ] ); rec( a := 1, b := 2, c := 3 ) gap> OptionRecordWithDefaults( defaults, [ rec( c := 8 ) ] ); rec( a := 1, b := 2, c := 8 ) gap> OptionRecordWithDefaults( defaults, rec( d := 9 ) ); Error, Unknown option: d gap> OptionRecordWithDefaults( defaults, [ rec( b := 7 ), rec( c := 8 ) ] ); Error, Too many arguments for function gap> OptionRecordWithDefaults( defaults, [6,7,8] ); Error, Too many arguments for function ]]> This function is designed to support functions with optional arguments given as a variable record, of the form function(x,y,options...). In the following, very contrived, example function, PrintDimensions, the defaults are given by the variable order which takes values h, w and d having default values 1, 2 and 3. If there is a second argument, then OptionRecordWithDefaults( order, arg[2] ); is used to cvhange the values. These three values then determine the order in which the three dimensions are printed using a SortParallel command.

1 ) and IsRecord( arg[2] ) then order := OptionRecordWithDefaults( order, arg[2] ); fi; L := [ order!.h, order!.w, order!.d ]; len := Length( L ); K := [ 1..len ]; SortParallel( L, K ); Print( "dimensions: " ); Print( V[K[1]], " = ", dim[K[1]], ", " ); Print( V[K[2]], " = ", dim[K[2]], ", " ); Print( V[K[3]], " = ", dim[K[3]], "\n" ); end;; ]]> In the example below the first call to PrintDimensions has just one parameter, mydim, so the default order is used. In the second call, alternate values for h, w and d are given, causing the width to be printed first, and then the depth and height. mydim := rec( height := 45, width := 31, depth := 17 ); rec( depth := 17, height := 45, width := 31 ) gap> PrintDimensions( mydim ); dimensions: height = 45, width = 31, depth = 17 gap> PrintDimensions( mydim, rec( h:=3, w:=1, d:=2 ) ); dimensions: width = 31, depth = 17, height = 45 ]]>
utils-0.86/doc/_entities.xml000644 000765 000024 00000000530 14621610512 021161 0ustar00christopherwensleystaff000000 000000 AutoDoc'> {\sf MathJax}'> ResClasses'> RCWA'> XMod'> Home'> {\sf Magma}'> Utils'> utils-0.86/doc/chap11.html000644 000765 000024 00000017337 14621610515 020437 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Chapter 11: Obsolete functions
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11 Obsolete functions
 11.1 Operations from AutoDoc
 11.2 Functions for printing
 11.3 Other obsolete functions

  11.3-1 Applicable Methods
  11.3-2 ExponentOfPrime

11 Obsolete functions

11.1 Operations from AutoDoc

The file functions FindMatchingFiles and CreateDirIfMissing were copied from package AutoDoc where they are named AutoDoc_FindMatchingFiles and AutoDoc_CreateDirIfMissing.

The string function StringDotSuffix was also copied from package AutoDoc, where it is named AUTODOC_GetSuffix.

The function SetIfMissing was also transferred from package AutoDoc, where it is called AUTODOC_SetIfMissing. It writes into a record provided the position is not yet bound.

As from version 0.61, all these functions became obsolete in Utils, but continue to be defined in AutoDoc.

11.2 Functions for printing

The function PrintOneItemPerLine was used to prints lists vertically, rather than horizontally. Since a very similar result may be achieved using the GAP library functions Perform and Display, this function became obsolete in version 0.61.


gap> s3 := SymmetricGroup( 3 );; 
gap> L := KnownPropertiesOfObject( GeneratorsOfGroup( s3 ) );;
gap> Perform( L, Display );
IsFinite
IsSmallList
IsGeneratorsOfMagmaWithInverses
IsGeneratorsOfSemigroup
IsSubsetLocallyFiniteGroup
gap> Perform( s3, Display ); 
()
(2,3)
(1,3)
(1,3,2)
(1,2,3)
(1,2)

11.3 Other obsolete functions

11.3-1 Applicable Methods

The function PrintApplicableMethod, which was included in versions from 0.41 to 0.58, has been removed since it was considered superfluous. The example shows how to print out a function.


gap> ApplicableMethod( IsCyclic, [ Group((1,2,3),(4,5)) ], 1, 1 );
#I  Searching Method for IsCyclic with 1 arguments:
#I  Total: 7 entries
#I  Method 4: ``IsCyclic'' at /Applications/gap/gap4r9/lib/grp.gi:30 , value: 
36
function( G ) ... end
gap> Print( last );
function ( G )
    if Length( GeneratorsOfGroup( G ) ) = 1 then
        return true;
    else
        TryNextMethod();
    fi;
    return;
end
gap> ApplicableMethod( IsCyclic, [ Group((1,2,3),(4,5)) ], 0, 3 );
function( <1 unnamed arguments> ) ... end
gap> Print( last );                                               
function ( <<arg-1>> )
    <<compiled GAP code from GAPROOT/lib/oper1.g:578>>
end

11.3-2 ExponentOfPrime

The function ExponentOfPrime was originally transferred from package RCWA. The command ExponentOfPrime(n,p) returned the exponent of the prime p in the prime factorization of n.

Since the GAP function PValuation produces the same results, and does so more quickly, this function has been made obsolete.

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utils-0.86/doc/chap8.txt000644 000765 000024 00000013625 14621610512 020231 0ustar00christopherwensleystaff000000 000000 8 Records 8.1 Functions for records 8.1-1 AssignGlobals AssignGlobals( rec )  function This function has been transferred from package RCWA. It assigns the record components of rec to global variables with the same names.  Example   gap> r := rec( a := 1, b := 2, c := 3 );;  gap> AssignGlobals( r ); The following global variables have been assigned: [ "a", "b", "c" ] gap> [a,b,c]; [ 1, 2, 3 ]   8.2 Option records for functions 8.2-1 OptionRecordWithDefaults OptionRecordWithDefaults( defaults, useroptions )  function This functions has been transferred by Chris Jefferson from other packages. It simplifies the handling of records which are intended to be used for expressing configuration options. defaults represents the "default record", and useroptions lets the user give new values for values in defaults. The function returns a record with the same component names as defaults and which has the same values as defaults, except for those component names in useroptions, where the values in useroptions are used instead. An error is given if useroptions contains any component names not in defaults. If useroptions is an empty list it is treated as an empty record, and if useroptions is a list of length 1 containing a record, this record is used as useroptions.  Example   gap> defaults := rec( a := 1, b := 2, c := 3 );; gap> OptionRecordWithDefaults( defaults, rec( a := 6) ); rec( a := 6, b := 2, c := 3 ) gap> OptionRecordWithDefaults( defaults, rec( b := 7, c := 8 ) ); rec( a := 1, b := 7, c := 8 ) gap> OptionRecordWithDefaults( defaults, [ ] ); rec( a := 1, b := 2, c := 3 ) gap> OptionRecordWithDefaults( defaults, [ rec( c := 8 ) ] ); rec( a := 1, b := 2, c := 8 ) gap> OptionRecordWithDefaults( defaults, rec( d := 9 ) ); Error, Unknown option: d gap> OptionRecordWithDefaults( defaults, [ rec( b := 7 ), rec( c := 8 ) ] ); Error, Too many arguments for function gap> OptionRecordWithDefaults( defaults, [6,7,8] ); Error, Too many arguments for function   This function is designed to support functions with optional arguments given as a variable record, of the form function(x,y,options...). In the following, very contrived, example function, PrintDimensions, the defaults are given by the variable order which takes values h, w and d having default values 1, 2 and 3. If there is a second argument, then OptionRecordWithDefaults( order, arg[2] ); is used to cvhange the values. These three values then determine the order in which the three dimensions are printed using a SortParallel command.   PrintDimensions := function( arg )   local nargs, dim, order, V, L, len, K, i;   nargs := Length( arg );   dim := [ arg[1]!.height, arg[1]!.width, arg[1]!.depth ];   order := rec( h := 1, w := 2, d := 3 );   V := [ "height", "width", "depth" ];   if ( nargs > 1 ) and IsRecord( arg[2] ) then   order := OptionRecordWithDefaults( order, arg[2] );   fi;   L := [ order!.h, order!.w, order!.d ];   len := Length( L );  K := [ 1..len ];   SortParallel( L, K );   Print( "dimensions: " );   Print( V[K[1]], " = ", dim[K[1]], ", " );  Print( V[K[2]], " = ", dim[K[2]], ", " );  Print( V[K[3]], " = ", dim[K[3]], "\n" ); end;;  In the example below the first call to PrintDimensions has just one parameter, mydim, so the default order is used. In the second call, alternate values for h, w and d are given, causing the width to be printed first, and then the depth and height.  Example   gap> mydim := rec( height := 45, width := 31, depth := 17 );  rec( depth := 17, height := 45, width := 31 ) gap> PrintDimensions( mydim ); dimensions: height = 45, width = 31, depth = 17 gap> PrintDimensions( mydim, rec( h:=3, w:=1, d:=2 ) ); dimensions: width = 31, depth = 17, height = 45   utils-0.86/doc/chap0_mj.html000644 000765 000024 00000046402 14621610515 021036 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Contents
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utils

Utility functions in GAP

0.86

16 May 2024

Thomas Breuer
Email: sam@math.rwth-aachen.de
Homepage: https://www.math.rwth-aachen.de/~Thomas.Breuer

Sebastian Gutsche
Email: gutsche@mathematik.uni-siegen.de
Homepage: https://sebasguts.github.io/

Max Horn
Email: mhorn@rptu.de
Homepage: https://github.com/mhorn

Alexander Hulpke
Email: hulpke@math.colostate.edu
Homepage: https://www.math.colostate.edu/~hulpke

Pedro García-Sánchez
Email: pedro@ugr.es
Homepage: http://www.ugr.es/local/pedro

Christopher Jefferson
Email: caj21@st-andrews.ac.uk
Homepage: https://caj.host.cs.st-andrews.ac.uk/

Stefan Kohl
Email: stefan@mcs.st-and.ac.uk
Homepage: https://www.gap-system.org/DevelopersPages/StefanKohl/

Frank Lübeck
Email: Frank.Luebeck@Math.RWTH-Aachen.De
Homepage: https://www.math.rwth-aachen.de/~Frank.Luebeck

Chris Wensley
Email: cdwensley.maths@btinternet.com
Homepage: https://github.com/cdwensley

Abstract

The Utils package provides a space for utility functions in a variety of GAP packages to be collected together into a single package. In this way it is hoped that they will become more visible to package authors.

Any package author who transfers a function to Utils will become an author of Utils.

If deemed appropriate, functions may also be transferred from the main library.

Bug reports, suggestions and comments are, of course, welcome. Please contact the last author at cdwensley.maths@btinternet.com or submit an issue at the GitHub repository https://github.com/gap-packages/utils/issues/.

Copyright

© 2015-2024, The GAP Group.

The Utils package is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

Acknowledgements

This documentation was prepared using the GAPDoc [LN17] and AutoDoc [GH16] packages.

The procedure used to produce new releases uses the package GitHubPagesForGAP [Hor17] and the package ReleaseTools.

Contents

1 Introduction
 1.1 Information for package authors
2 Printing Lists and Iterators
 2.1 Printing selected items

  2.1-1 PrintSelection
3 Lists, Sets and Strings
 3.1 Functions for lists

  3.1-1 DifferencesList
  3.1-2 QuotientsList
  3.1-3 SearchCycle
  3.1-4 RandomCombination
 3.2 Distinct and Common Representatives

  3.2-1 DistinctRepresentatives
 3.3 Functions for strings

  3.3-1 BlankFreeString
4 Number-theoretic functions
 4.1 Functions for integers

  4.1-1 AllSmoothIntegers
  4.1-2 AllProducts
  4.1-3 RestrictedPartitionsWithoutRepetitions
  4.1-4 NextProbablyPrimeInt
  4.1-5 PrimeNumbersIterator
5 Groups and homomorphisms
 5.1 Functions for groups

  5.1-1 Comm
  5.1-2 IsCommuting
  5.1-3 ListOfPowers
  5.1-4 GeneratorsAndInverses
  5.1-5 UpperFittingSeries
 5.2 Left Cosets for Groups

  5.2-1 LeftCoset
  5.2-2 Inverse
 5.3 Functions for group homomorphisms

  5.3-1 EpimorphismByGenerators
  5.3-2 Pullback
  5.3-3 CentralProduct
  5.3-4 IdempotentEndomorphisms
  5.3-5 DirectProductOfFunctions
  5.3-6 DirectProductOfAutomorphismGroups
6 Matrices
 6.1 Some operations for matrices

  6.1-1 DirectSumDecompositionMatrices
7 Iterators
 7.1 Some iterators for groups and their isomorphisms

  7.1-1 AllIsomorphismsIterator
  7.1-2 AllSubgroupsIterator
 7.2 Operations on iterators

  7.2-1 CartesianIterator
  7.2-2 UnorderedPairsIterator
8 Records
 8.1 Functions for records

  8.1-1 AssignGlobals
 8.2 Option records for functions

  8.2-1 OptionRecordWithDefaults
9 Web Downloads
 9.1 Functions for downloading files from the web

  9.1-1 Download
  9.1-2 User preference DownloadVerifyCertificate
10 Various other functions
 10.1 File operations

  10.1-1 Log2HTML
 10.2 LaTeX strings

  10.2-1 IntOrOnfinityToLaTeX
  10.2-2 LaTeXStringFactorsInt
 10.3 Conversion to \({\sf Magma}\) string

  10.3-1 ConvertToMagmaInputString
11 Obsolete functions
 11.1 Operations from AutoDoc
 11.2 Functions for printing
 11.3 Other obsolete functions

  11.3-1 Applicable Methods
  11.3-2 ExponentOfPrime
12 The transfer procedure
References
Index

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utils-0.86/doc/chap9.txt000644 000765 000024 00000011710 14621610512 020223 0ustar00christopherwensleystaff000000 000000 9 Web Downloads The Download operation has been written by Thomas Breuer, incorporating a number of suggestions from Max Horn, for version 0.77 of Utils. It implements downloading a file from within GAP. It can use the IO or curlInterface packages, or wget or curl, if installed, and it can be extended with other download methods quite easily. It is envisaged that, once other packages have started to use it, and any problems have been addressed, that the functions will be transferred to the main GAP library. 9.1 Functions for downloading files from the web 9.1-1 Download Download( url[, opt] )  function This function downloads the file with the web address url, which must be a string. The result is a record which has at least the component success, with value true if the download was successful and false otherwise. In the former case, the component result is bound, whose value is a string that contains the contents of the downloaded file. In the latter case, the component error is bound, whose value is a string that describes the problem. The function calls the methods stored in the global list Download_Methods until one of them is successful. Currently there are methods based on the GAP functions DownloadURL (curl: DownloadURL) and SingleHTTPRequest (IO: SingleHTTPRequest), and methods based on the external programs wget and curl. An optional record opt can be given. The following components are supported. target If this component is bound then its value must be a string that is a local filename, and the function writes the downloaded contents to this file; the returned record does not have a result component in this case. verifyCert If this component is bound and has the value false then those download methods that are based on curl or wget will omit the check of the server's certificate. The same effect is achieved for all Download calls by setting the user preference DownloadVerifyCertificate (see 9.1-2) to false and omitting the verifyCert component from opt.  Example   gap> url:= "https://www.gap-system.org/Packages/utils.html";; gap> res1:= Download( url );; gap> res1.success; true gap> IsBound( res1.result ) and IsString( res1.result ); true gap> res2:= Download( Concatenation( url, "xxx" ) );; gap> res2.success; false gap> IsBound( res2.error ) and IsString( res2.error ); true   9.1-2 User preference DownloadVerifyCertificate The value true (the default) means that the server's certificate is checked in calls of Download (9.1-1), such that nothing gets downloaded if the certificate is invalid. If the value is false then download methods are supposed to omit the check of the server's certificate (this may not be supported by all download methods). One can set the value of the preference to be val via SetUserPreference (Reference: SetUserPreference), by calling SetUserPreference( "utils", "DownloadVerifyCertificate", val ), and access the current value via UserPreference (Reference: UserPreference), by calling UserPreference( "utils", "DownloadVerifyCertificate" ). We recommend leaving this preference at its default value true. Sometimes it can be necessary to change it, e.g. to work around issues with old operating systems which may not be able to correctly verify new certificates. In general it is better to update such a system, but if that is not an option, then disabling certificate checks may be a good last resort. utils-0.86/doc/chap2.html000644 000765 000024 00000013236 14621610515 020351 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Chapter 2: Printing Lists and Iterators
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2 Printing Lists and Iterators
 2.1 Printing selected items

  2.1-1 PrintSelection

2 Printing Lists and Iterators

2.1 Printing selected items

The functions described here print lists or objects with an iterator with one item per line, either the whole list/iterator or certain subsets:

  • by giving a list of positions of items to be printed, or

  • by specifying a first item and then a regular step.

2.1-1 PrintSelection
‣ PrintSelection( obj, first, step[, last] )( function )
‣ PrintSelection( obj, list )( function )

This function, given three (or four) parameters, calls operations PrintSelectionFromList or PrintSelectionFromIterator which prints the first item specified, and then the item at every step. The fourth parameter is essential when the object being printed is infinite.

Alternatively, given two parameters, with the second parameter a list L of positive integers, only the items at positions in L are printed.


gap> L := List( [1..20], n -> n^5 );;
gap> PrintSelection( L, [18..20] );
18 : 1889568
19 : 2476099
20 : 3200000
gap> PrintSelection( L, 2, 9 );       
2 : 32
11 : 161051
20 : 3200000
gap> PrintSelection( L, 2, 3, 11 );
2 : 32
5 : 3125
8 : 32768
11 : 161051
gap> s5 := SymmetricGroup( 5 );;
gap> PrintSelection( s5, [30,31,100,101] );
30 : (1,5)(3,4)
31 : (1,5,2)
100 : (1,4,3)
101 : (1,4)(3,5)
gap> PrintSelection( s5, 1, 30 );
1 : ()
31 : (1,5,2)
61 : (1,2,3)
91 : (1,3,5,2,4)
gap> PrintSelection( s5, 9, 11, 43 );
9 : (2,5,3)
20 : (2,4)
31 : (1,5,2)
42 : (1,5,2,3,4)
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utils-0.86/doc/chap3.html000644 000765 000024 00000040234 14621610515 020350 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Chapter 3: Lists, Sets and Strings
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3 Lists, Sets and Strings
 3.1 Functions for lists

  3.1-1 DifferencesList
  3.1-2 QuotientsList
  3.1-3 SearchCycle
  3.1-4 RandomCombination
 3.2 Distinct and Common Representatives

  3.2-1 DistinctRepresentatives
 3.3 Functions for strings

  3.3-1 BlankFreeString

3 Lists, Sets and Strings

3.1 Functions for lists

3.1-1 DifferencesList
‣ DifferencesList( L )( function )

This function has been transferred from package ResClasses.

It takes a list L of length n and outputs the list of length n-1 containing all the differences L[i]-L[i-1].


gap> List( [1..12], n->n^3 );
[ 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 ]
gap> DifferencesList( last );
[ 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397 ]
gap> DifferencesList( last );
[ 12, 18, 24, 30, 36, 42, 48, 54, 60, 66 ]
gap> DifferencesList( last );
[ 6, 6, 6, 6, 6, 6, 6, 6, 6 ]

3.1-2 QuotientsList
‣ QuotientsList( L )( function )
‣ FloatQuotientsList( L )( function )

These functions have been transferred from package ResClasses.

They take a list L of length n and output the quotients L[i]/L[i-1] of consecutive entries in L. An error is returned if an entry is zero.


gap> List( [0..10], n -> Factorial(n) );
[ 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 ]
gap> QuotientsList( last );
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]
gap> L := [ 1, 3, 5, -1, -3, -5 ];;
gap> QuotientsList( L );
[ 3, 5/3, -1/5, 3, 5/3 ]
gap> FloatQuotientsList( L );
[ 3., 1.66667, -0.2, 3., 1.66667 ]
gap> QuotientsList( [ 2, 1, 0, -1, -2 ] );
[ 1/2, 0, fail, 2 ]
gap> FloatQuotientsList( [1..10] );
[ 2., 1.5, 1.33333, 1.25, 1.2, 1.16667, 1.14286, 1.125, 1.11111 ]
gap> Product( last );
10.

3.1-3 SearchCycle
‣ SearchCycle( L )( operation )

This function has been transferred from package RCWA.

SearchCycle is a tool to find likely cycles in lists. What, precisely, a cycle is, is deliberately fuzzy here, and may possibly even change. The idea is that the beginning of the list may be anything, following that the same pattern needs to be repeated several times in order to be recognized as a cycle.


gap> L := [1..20];;  L[1]:=13;;                                              
gap> for i in [1..19] do                                                     
>        if IsOddInt(L[i]) then L[i+1]:=3*L[i]+1; else L[i+1]:=L[i]/2; fi;
>    od;                                                                  
gap> L;                                                                      
[ 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4 ]
gap> SearchCycle( L );                                                       
[ 1, 4, 2 ]
gap> n := 1;;  L := [n];;
gap> for i in [1..100] do  n:=(n^2+1) mod 1093;  Add(L,n);  od;
gap> L; 
[ 1, 2, 5, 26, 677, 363, 610, 481, 739, 715, 795, 272, 754, 157, 604, 848, 
  1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 
  211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 
  378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 
  754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 
  848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 
  271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004 ]
gap> C := SearchCycle( L );
[ 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754 ]
gap> P := Positions( L, 157 );
[ 14, 26, 38, 50, 62, 74, 86, 98 ]
gap> Length( C );  DifferencesList( P );
12
[ 12, 12, 12, 12, 12, 12, 12 ]

3.1-4 RandomCombination
‣ RandomCombination( S, k )( operation )

This function has been transferred from package ResClasses.

It returns a random unordered k-tuple of distinct elements of a set S.


gap> ## "6 aus 49" is a common lottery in Germany
gap> RandomCombination( [1..49], 6 ); 
[ 2, 16, 24, 26, 37, 47 ]

3.2 Distinct and Common Representatives

3.2-1 DistinctRepresentatives
‣ DistinctRepresentatives( list )( operation )
‣ CommonRepresentatives( list )( operation )
‣ CommonTransversal( grp, subgrp )( operation )
‣ IsCommonTransversal( grp, subgrp, list )( operation )

These operations have been transferred from package XMod.

They deal with lists of subsets of [1 ... n] and construct systems of distinct and common representatives using simple, non-recursive, combinatorial algorithms.

When L is a set of n subsets of [1 ... n] and the Hall condition is satisfied (the union of any k subsets has at least k elements), a set of DistinctRepresentatives exists.

When J,K are both lists of n sets, the operation CommonRepresentatives returns two lists: the set of representatives, and a permutation of the subsets of the second list.

The operation CommonTransversal may be used to provide a common transversal for the sets of left and right cosets of a subgroup H of a group G, although a greedy algorithm is usually quicker.


gap> J := [ [1,2,3], [3,4], [3,4], [1,2,4] ];;
gap> DistinctRepresentatives( J );
[ 1, 3, 4, 2 ]
gap> K := [ [3,4], [1,2], [2,3], [2,3,4] ];;
gap> CommonRepresentatives( J, K );
[ [ 3, 3, 3, 1 ], [ 1, 3, 4, 2 ] ]
gap> d16 := DihedralGroup( IsPermGroup, 16 ); 
Group([ (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) ])
gap> SetName( d16, "d16" );
gap> c4 := Subgroup( d16, [ d16.1^2 ] ); 
Group([ (1,3,5,7)(2,4,6,8) ])
gap> SetName( c4, "c4" );
gap> RightCosets( d16, c4 );
[ RightCoset(c4,()), RightCoset(c4,(2,8)(3,7)(4,6)), RightCoset(c4,(1,8,7,6,5,
   4,3,2)), RightCoset(c4,(1,8)(2,7)(3,6)(4,5)) ]
gap> trans := CommonTransversal( d16, c4 );
[ (), (2,8)(3,7)(4,6), (1,2,3,4,5,6,7,8), (1,2)(3,8)(4,7)(5,6) ]
gap> IsCommonTransversal( d16, c4, trans );
true

3.3 Functions for strings

3.3-1 BlankFreeString
‣ BlankFreeString( obj )( function )

This function has been transferred from package ResClasses.

The result of BlankFreeString( obj ); is a composite of the functions String( obj ) and RemoveCharacters( obj, " " );.


gap> gens := GeneratorsOfGroup( DihedralGroup(12) );
[ f1, f2, f3 ]
gap> String( gens );                                
"[ f1, f2, f3 ]"
gap> BlankFreeString( gens );                       
"[f1,f2,f3]"
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utils-0.86/doc/chap1_mj.html000644 000765 000024 00000021360 14621610515 021033 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Chapter 1: Introduction
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1 Introduction
 1.1 Information for package authors

1 Introduction

The Utils package provides a space for utility functions from a variety of GAP packages to be collected together into a single package. In this way it is hoped that they will become more visible to other package authors. This package was first distributed as part of the GAP 4.8.2 distribution.

The package is loaded with the command


gap> LoadPackage( "utils" );

Functions have been transferred from the following packages:

  • Conversion of a GAP group to a \({\sf Magma}\) output string, taken from various sources including other.gi in the main library.

Transfer is complete (for now) for functions from the following packages:

The package may be obtained either as a compressed .tar file or as a .zip file, utils-version_number.tar.gz, by ftp from one of the following sites:

The package also has a GitHub repository at: https://github.com/gap-packages/utils.

Once the package is loaded, the manual doc/manual.pdf can be found in the documentation folder. The html versions, with or without \({\sf MathJax}\), may be rebuilt as follows:


gap> ReadPackage( "utils", "makedoc.g" );

It is possible to check that the package has been installed correctly by running the test files (which terminates the GAP session):


gap> ReadPackage( "utils", "tst/testall.g" );
Architecture: . . . . . 
testing: . . . . . 
. . . 
#I  No errors detected while testing

Note that functions listed in this manual that are currently in the process of being transferred are only read from the source package Home (say), and so can only be used if Home has already been loaded. There are no such functions in transition at present.

1.1 Information for package authors

A function (or collection of functions) is suitable for transfer from a package Home to Utils if the following conditions are satisfied.

  • The function is sufficiently non-specialised so that it might be of use to other authors.

  • The function does not depend on the remaining functions in Home

  • The function does not do what can already be done with a GAP library function.

  • Documentation of the function and test examples are available.

  • When there is more than one active author of Home, they should all be aware (and content) that the transfer is taking place.

Authors of packages may be reluctant to let go of their utility functions. The following principles may help to reassure them. (Suggestions for more items here are welcome.)

  • A function that has been transferred to Utils will not be changed without the approval of the original author.

  • The current package maintainer has every intention of continuing to maintain Utils. In the event that this proves impossible, the GAP development team will surely find someone to take over.

  • Function names will not be changed unless specifically requested by Home's author(s) or unless they have the form HOME_FunctionName.

  • In order to speed up the transfer process, only functions from one package will be in transition at any given time. Hopefully a week or two will suffice for most packages.

  • Any package author who transfers a function to Utils will become an author of Utils. (In truth, Utils does not have authors, just a large number of contributors.)

The process for transferring utility functions from Home to Utils is described in Chapter 12.

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utils-0.86/doc/title.xml000644 000765 000024 00000006140 14621610512 020322 0ustar00christopherwensleystaff000000 000000 utils Utility functions in &GAP; 0.86 Thomas Breuer
 sam@math.rwth-aachen.de https://www.math.rwth-aachen.de/~Thomas.Breuer Sebastian Gutsche
 gutsche@mathematik.uni-siegen.de https://sebasguts.github.io/ Max Horn
 mhorn@rptu.de https://github.com/mhorn Alexander Hulpke
 hulpke@math.colostate.edu https://www.math.colostate.edu/~hulpke Pedro García-Sánchez
 pedro@ugr.es http://www.ugr.es/local/pedro Christopher Jefferson
 caj21@st-andrews.ac.uk https://caj.host.cs.st-andrews.ac.uk/ Stefan Kohl
 stefan@mcs.st-and.ac.uk https://www.gap-system.org/DevelopersPages/StefanKohl/ Frank Lübeck
 Frank.Luebeck@Math.RWTH-Aachen.De https://www.math.rwth-aachen.de/~Frank.Luebeck Chris Wensley
 cdwensley.maths@btinternet.com https://github.com/cdwensley 16 May 2024 The &Utils; package provides a space for utility functions in a variety of &GAP; packages to be collected together into a single package. In this way it is hoped that they will become more visible to package authors.

Any package author who transfers a function to &Utils; will become an author of &Utils;.

If deemed appropriate, functions may also be transferred from the main library.

Bug reports, suggestions and comments are, of course, welcome. Please contact the last author at cdwensley.maths@btinternet.com or submit an issue at the GitHub repository https://github.com/gap-packages/utils/issues/. ©right; 2015-2024, The GAP Group.

The &Utils; package is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This documentation was prepared using the &GAPDoc; and &AutoDoc; packages.

The procedure used to produce new releases uses the package GitHubPagesForGAP and the package ReleaseTools.

utils-0.86/doc/download.xml000644 000765 000024 00000011330 14621444241 021011 0ustar00christopherwensleystaff000000 000000 Web Downloads The Download operation has been written by Thomas Breuer, incorporating a number of suggestions from Max Horn, for version 0.77 of Utils. It implements downloading a file from within &GAP;. It can use the IO or curlInterface packages, or wget or curl, if installed, and it can be extended with other download methods quite easily. It is envisaged that, once other packages have started to use it, and any problems have been addressed, that the functions will be transferred to the main &GAP; library.

Functions for downloading files from the web This function downloads the file with the web address url, which must be a string.

The result is a record which has at least the component success, with value true if the download was successful and false otherwise. In the former case, the component result is bound, whose value is a string that contains the contents of the downloaded file. In the latter case, the component error is bound, whose value is a string that describes the problem.

The function calls the methods stored in the global list Download_Methods until one of them is successful. Currently there are methods based on the &GAP; functions and , and methods based on the external programs wget and curl.

An optional record opt can be given. The following components are supported.

target If this component is bound then its value must be a string that is a local filename, and the function writes the downloaded contents to this file; the returned record does not have a result component in this case. verifyCert If this component is bound and has the value false then those download methods that are based on curl or wget will omit the check of the server's certificate. The same effect is achieved for all calls by setting the user preference DownloadVerifyCertificate (see ) to false and omitting the verifyCert component from opt.

url:= "https://www.gap-system.org/Packages/utils.html";; gap> res1:= Download( url );; gap> res1.success; true gap> IsBound( res1.result ) and IsString( res1.result ); true gap> res2:= Download( Concatenation( url, "xxx" ) );; gap> res2.success; false gap> IsBound( res2.error ) and IsString( res2.error ); true ]]> User preference DownloadVerifyCertificate DownloadVerifyCertificate The value true (the default) means that the server's certificate is checked in calls of , such that nothing gets downloaded if the certificate is invalid.

If the value is false then download methods are supposed to omit the check of the server's certificate (this may not be supported by all download methods).

One can set the value of the preference to be val via , by calling SetUserPreference( "utils", "DownloadVerifyCertificate", val ), and access the current value via , by calling UserPreference( "utils", "DownloadVerifyCertificate" ).

We recommend leaving this preference at its default value true. Sometimes it can be necessary to change it, e.g. to work around issues with old operating systems which may not be able to correctly verify new certificates. In general it is better to update such a system, but if that is not an option, then disabling certificate checks may be a good last resort.

utils-0.86/doc/manual.js000644 000765 000024 00000010113 14621610515 020270 0ustar00christopherwensleystaff000000 000000 /* manual.js Frank Lübeck */ /* This file contains a few javascript functions which allow to switch between display styles for GAPDoc HTML manuals. If javascript is switched off in a browser or this file in not available in a manual directory, this is no problem. Users just cannot switch between several styles and don't see the corresponding button. A style with name mystyle can be added by providing two files (or only one of them). mystyle.js: Additional javascript code for the style, it is read in the HTML pages after this current file. The additional code may adjust the preprocessing function jscontent() with is called onload of a file. This is done by appending functions to jscontentfuncs (jscontentfuncs.push(newfunc);). Make sure, that your style is still usable without javascript. mystyle.css: CSS configuration, read after manual.css (so it can just reconfigure a few details, or overwrite everything). Then adjust chooser.html such that users can switch on and off mystyle. A user can change the preferred style permanently by using the [Style] link and choosing one. Or one can append '?GAPDocStyle=mystyle' to the URL when loading any file of the manual (so the style can be configured in the GAP user preferences). */ /* generic helper function */ function deleteCookie(nam) { document.cookie = nam+"=;Path=/;expires=Thu, 01 Jan 1970 00:00:00 GMT"; } /* read a value from a "nam1=val1;nam2=val2;..." string (e.g., the search part of an URL or a cookie */ function valueString(str,nam) { var cs = str.split(";"); for (var i=0; i < cs.length; i++) { var pos = cs[i].search(nam+"="); if (pos > -1) { pos = cs[i].indexOf("="); return cs[i].slice(pos+1); } } return 0; } /* when a non-default style is chosen via URL or a cookie, then the cookie is reset and the styles .js and .css files are read */ function overwriteStyle() { /* style in URL? */ var style = valueString(window.location.search, "GAPDocStyle"); /* otherwise check cookie */ if (style == 0) style = valueString(document.cookie, "GAPDocStyle"); if (style == 0) return; if (style == "default") deleteCookie("GAPDocStyle"); else { /* ok, we set the cookie for path "/" */ var path = "/"; /* or better like this ??? var here = window.location.pathname.split("/"); for (var i=0; i+3 < here.length; i++) path = path+"/"+here[i]; */ document.cookie = "GAPDocStyle="+style+";Path="+path; /* split into names of style files */ var stlist = style.split(","); /* read style's css and js files */ for (var i=0; i < stlist.length; i++) { document.writeln(''); document.writeln(''); } } } /* this adds a "[Style]" link next to the MathJax switcher */ function addStyleLink() { var line = document.getElementById("mathjaxlink"); var el = document.createElement("a"); var oncl = document.createAttribute("href"); var back = window.location.protocol+"//" if (window.location.protocol == "http:" || window.location.protocol == "https:") { back = back+window.location.host; if (window.location.port != "") { back = back+":"+window.location.port; } } back = back+window.location.pathname; oncl.nodeValue = "chooser.html?BACK="+back; el.setAttributeNode(oncl); var cont = document.createTextNode(" [Style]"); el.appendChild(cont); line.appendChild(el); } var jscontentfuncs = new Array(); jscontentfuncs.push(addStyleLink); /* the default jscontent() only adds the [Style] link to the page */ function jscontent () { for (var i=0; i < jscontentfuncs.length; i++) jscontentfuncs[i](); } utils-0.86/doc/number.xml000644 000765 000024 00000013234 14621444241 020477 0ustar00christopherwensleystaff000000 000000 Number-theoretic functions
Functions for integers This function has been transferred from package &RCWA;.

smooth integer The function AllSmoothIntegers(maxp,maxn) returns the list of all positive integers less than or equal to maxn whose prime factors are all in the list L = \{p ~|~ p \leqslant maxp, p~\mbox{prime} \}.

In the alternative form, when L is a list of primes, the function returns the list of all positive integers whose prime factors lie in L.

AllSmoothIntegers( 3, 1000 ); [ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576, 648, 729, 768, 864, 972 ] gap> AllSmoothIntegers( [5,11,17], 1000 ); [ 1, 5, 11, 17, 25, 55, 85, 121, 125, 187, 275, 289, 425, 605, 625, 935 ] gap> Length( last ); 16 gap> List( [3..20], n -> Length( AllSmoothIntegers( [5,11,17], 10^n ) ) ); [ 16, 29, 50, 78, 114, 155, 212, 282, 359, 452, 565, 691, 831, 992, 1173, 1374, 1595, 1843 ] ]]> This function has been transferred from package &RCWA;.

The command AllProducts(L,k) returns the list of all products of k entries of the list L. Note that every ordering of the entries is used so that, in the commuting case, there are bound to be repetitions.

AllProducts([1..4],3); [ 1, 2, 3, 4, 2, 4, 6, 8, 3, 6, 9, 12, 4, 8, 12, 16, 2, 4, 6, 8, 4, 8, 12, 16, 6, 12, 18, 24, 8, 16, 24, 32, 3, 6, 9, 12, 6, 12, 18, 24, 9, 18, 27, 36, 12, 24, 36, 48, 4, 8, 12, 16, 8, 16, 24, 32, 12, 24, 36, 48, 16, 32, 48, 64 ] gap> Set(last); [ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 64 ] gap> AllProducts( [(1,2,3),(2,3,4)], 2 ); [ (2,4,3), (1,2)(3,4), (1,3)(2,4), (1,3,2) ] ]]> This function has been transferred from package &RCWA;.

For a positive integer n and a set of positive integers S, this function returns the list of partitions of n into distinct elements of S. Unlike RestrictedPartitions, no repetitions are allowed.

RestrictedPartitions( 20, [4..10] ); [ [ 4, 4, 4, 4, 4 ], [ 5, 5, 5, 5 ], [ 6, 5, 5, 4 ], [ 6, 6, 4, 4 ], [ 7, 5, 4, 4 ], [ 7, 7, 6 ], [ 8, 4, 4, 4 ], [ 8, 6, 6 ], [ 8, 7, 5 ], [ 8, 8, 4 ], [ 9, 6, 5 ], [ 9, 7, 4 ], [ 10, 5, 5 ], [ 10, 6, 4 ], [ 10, 10 ] ] gap> RestrictedPartitionsWithoutRepetitions( 20, [4..10] ); [ [ 10, 6, 4 ], [ 9, 7, 4 ], [ 9, 6, 5 ], [ 8, 7, 5 ] ] gap> RestrictedPartitionsWithoutRepetitions( 10^2, List([1..10], n->n^2 ) ); [ [ 100 ], [ 64, 36 ], [ 49, 25, 16, 9, 1 ] ] ]]> This function has been transferred from package &RCWA;.

The function NextProbablyPrimeInt(n) does the same as NextPrimeInt(n) except that for reasons of performance it tests numbers only for IsProbablyPrimeInt(n) instead of IsPrimeInt(n). For large n, this function is much faster than NextPrimeInt(n)

n := 2^251; 3618502788666131106986593281521497120414687020801267626233049500247285301248 gap> NextProbablyPrimeInt( n ); 3618502788666131106986593281521497120414687020801267626233049500247285301313 gap> time; 1 gap> NextPrimeInt( n ); 3618502788666131106986593281521497120414687020801267626233049500247285301313 gap> time; 213 ]]> This function has been transferred from package &RCWA;.

This function returns an iterator which runs over the prime numbers n ascending order; it takes an optional argument chunksize which specifies the length of the interval which is sieved in one go (the default is 10^7), and which can be used to balance runtime vs. memory consumption. It is assumed that chunksize is larger than any gap between two consecutive primes within the range one intends to run the iterator over.

iter := PrimeNumbersIterator();; gap> for i in [1..100] do p := NextIterator(iter); od; gap> p; 541 gap> sum := 0;; gap> ## "prime number race" 1 vs. 3 mod 4 gap> for p in PrimeNumbersIterator() do > if p <> 2 then sum := sum + E(4)^(p-1); fi; > if sum > 0 then break; fi; > od; gap> p; 26861 ]]>

utils-0.86/doc/chap10.html000644 000765 000024 00000024624 14621610515 020433 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Chapter 10: Various other functions
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10 Various other functions
 10.1 File operations

  10.1-1 Log2HTML
 10.2 LaTeX strings

  10.2-1 IntOrOnfinityToLaTeX
  10.2-2 LaTeXStringFactorsInt
 10.3 Conversion to Magma string

  10.3-1 ConvertToMagmaInputString

10 Various other functions

10.1 File operations

10.1-1 Log2HTML
‣ Log2HTML( filename )( function )

This function has been transferred from package RCWA.

This function converts the GAP logfile filename to HTML. It appears that the logfile should be in your current directory. The extension of the input file must be *.log. The name of the output file is the same as the one of the input file except that the extension *.log is replaced by *.html. There is a sample CSS file in utils/doc/gaplog.css, which you can adjust to your taste.


gap> LogTo( "triv.log" );
gap> a := 33^5;
39135393
gap> LogTo(); 
gap> Log2HTML( "triv.log" );

10.2 LaTeX strings

10.2-1 IntOrOnfinityToLaTeX
‣ IntOrOnfinityToLaTeX( n )( function )

This function has been transferred from package ResClasses.

IntOrInfinityToLaTeX(n) returns the LaTeX string for n.


gap> IntOrInfinityToLaTeX( 10^3 );
"1000"
gap> IntOrInfinityToLaTeX( infinity );
"\\infty"

10.2-2 LaTeXStringFactorsInt
‣ LaTeXStringFactorsInt( n )( function )

This function has been transferred from package RCWA.

It returns the prime factorization of the integer n as a string in LaTeX format.


gap> LaTeXStringFactorsInt( Factorial(12) );
"2^{10} \\cdot 3^5 \\cdot 5^2 \\cdot 7 \\cdot 11"

10.3 Conversion to Magma string

10.3-1 ConvertToMagmaInputString
‣ ConvertToMagmaInputString( arg )( function )

The function ConvertToMagmaInputString( obj [, str] ) attempts to output a string s which can be read into Magma [BCP97] so as to produce the same group in that computer algebra system. In the second form the user specifies the name of the resulting object, so that the output string has the form "str := ...". When obj is a permutation group, the operation PermGroupToMagmaFormat(obj) is called. This function has been taken from other.gi in the main library where it was called MagmaInputString. When obj is a pc-group, the operation PcGroupToMagmaFormat(obj) is called. This function was private code of Max Horn. When obj is a matrix group over a finite field, the operation MatrixGroupToMagmaFormat(obj) is called. This function is a modification of private code of Frank Lübeck.

Hopefully code for other types of group will be added in due course.

These functions should be considered experimental, and more testing is desirable.


gap> ConvertToMagmaInputString( Group( (1,2,3,4,5), (3,4,5) ) );
"PermutationGroup<5|(1,2,3,4,5),\n(3,4,5)>;\n"
gap> ConvertToMagmaInputString( Group( (1,2,3,4,5) ), "c5" );        
"c5:=PermutationGroup<5|(1,2,3,4,5)>;\n"
gap> ConvertToMagmaInputString( DihedralGroup( IsPcGroup, 10 ) );
"PolycyclicGroup< f1,f2 |\nf1^2,\nf2^5,\nf2^f1 = f2^4\n>;\n"
gap> M := GL(2,5);;  Size(M); 
480
gap> s1 := ConvertToMagmaInputString( M );
"F := GF(5);\nP := GL(2,F);\ngens := [\nP![2,0,0,1],\nP![4,1,4,0]\n];\nsub<P |\
 gens>;\n"
gap> Print( s1 );
F := GF(5);
P := GL(2,F);
gens := [
P![2,0,0,1],
P![4,1,4,0]
];
sub<P | gens>;
gap> n1 := [ [ Z(9)^0, Z(9)^0 ], [ Z(9)^0, Z(9) ] ];;
gap> n2 := [ [ Z(9)^0, Z(9)^3 ], [ Z(9)^4, Z(9)^2 ] ];;
gap> N := Group( n1, n2 );;  Size( N );
5760
gap> s2 := ConvertToMagmaInputString( N, "gpN" );;
gap> Print( s2 );
F := GF(3^2);
P := GL(2,F);
w := PrimitiveElement(F);
gens := [
P![ 1, 1, 1,w^1],
P![ 1,w^3, 2,w^2]
];
gpN := sub<P | gens>;
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8 Records
 8.1 Functions for records

  8.1-1 AssignGlobals
 8.2 Option records for functions

  8.2-1 OptionRecordWithDefaults

8 Records

8.1 Functions for records

8.1-1 AssignGlobals
‣ AssignGlobals( rec )( function )

This function has been transferred from package RCWA.

It assigns the record components of rec to global variables with the same names.


gap> r := rec( a := 1, b := 2, c := 3 );;                                      
gap> AssignGlobals( r );
The following global variables have been assigned:
[ "a", "b", "c" ]
gap> [a,b,c];
[ 1, 2, 3 ]

8.2 Option records for functions

8.2-1 OptionRecordWithDefaults
‣ OptionRecordWithDefaults( defaults, useroptions )( function )

This functions has been transferred by Chris Jefferson from other packages. It simplifies the handling of records which are intended to be used for expressing configuration options. defaults represents the "default record", and useroptions lets the user give new values for values in defaults.

The function returns a record with the same component names as defaults and which has the same values as defaults, except for those component names in useroptions, where the values in useroptions are used instead. An error is given if useroptions contains any component names not in defaults. If useroptions is an empty list it is treated as an empty record, and if useroptions is a list of length \(1\) containing a record, this record is used as useroptions.


gap> defaults := rec( a := 1, b := 2, c := 3 );;
gap> OptionRecordWithDefaults( defaults, rec( a := 6) );
rec( a := 6, b := 2, c := 3 )
gap> OptionRecordWithDefaults( defaults, rec( b := 7, c := 8 ) );
rec( a := 1, b := 7, c := 8 )
gap> OptionRecordWithDefaults( defaults, [ ] );
rec( a := 1, b := 2, c := 3 )
gap> OptionRecordWithDefaults( defaults, [ rec( c := 8 ) ] );
rec( a := 1, b := 2, c := 8 )
gap> OptionRecordWithDefaults( defaults, rec( d := 9 ) );
Error, Unknown option: d
gap> OptionRecordWithDefaults( defaults, [ rec( b := 7 ), rec( c := 8 ) ] );
Error, Too many arguments for function
gap> OptionRecordWithDefaults( defaults, [6,7,8] );
Error, Too many arguments for function

This function is designed to support functions with optional arguments given as a variable record, of the form function(x,y,options...). In the following, very contrived, example function, PrintDimensions, the defaults are given by the variable order which takes values h, w and d having default values \(1\), \(2\) and \(3\). If there is a second argument, then OptionRecordWithDefaults( order, arg[2] ); is used to cvhange the values. These three values then determine the order in which the three dimensions are printed using a SortParallel command.


PrintDimensions := function( arg ) 
    local nargs, dim, order, V, L, len, K, i; 
    nargs := Length( arg ); 
    dim := [ arg[1]!.height, arg[1]!.width, arg[1]!.depth ]; 
    order := rec( h := 1, w := 2, d := 3 ); 
    V := [ "height", "width", "depth" ]; 
    if ( nargs > 1 ) and IsRecord( arg[2] ) then 
        order := OptionRecordWithDefaults( order, arg[2] ); 
    fi; 
    L := [ order!.h, order!.w, order!.d ]; 
    len := Length( L );
    K := [ 1..len ]; 
    SortParallel( L, K ); 
    Print( "dimensions: " ); 
    Print( V[K[1]], " = ", dim[K[1]], ", " );
    Print( V[K[2]], " = ", dim[K[2]], ", " );
    Print( V[K[3]], " = ", dim[K[3]], "\n" );
end;;

In the example below the first call to PrintDimensions has just one parameter, mydim, so the default order is used. In the second call, alternate values for h, w and d are given, causing the width to be printed first, and then the depth and height.


gap> mydim := rec( height := 45, width := 31, depth := 17 ); 
rec( depth := 17, height := 45, width := 31 )
gap> PrintDimensions( mydim );
dimensions: height = 45, width = 31, depth = 17
gap> PrintDimensions( mydim, rec( h:=3, w:=1, d:=2 ) );
dimensions: width = 31, depth = 17, height = 45
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utils-0.86/doc/bib.xml.bib000644 000765 000024 00000011417 14621610512 020473 0ustar00christopherwensleystaff000000 000000 @manual{ AutoDoc, author = {Gutsche, S. and Horn, M.}, title = {{AutoDoc \texttt{\symbol{45}} Generate documentation from GAP source code (Version 2016.12.04)}}, year = {2016}, note = {GAP package, \href {https://github.com/gap-packages/AutoDoc} {\texttt{https://github.com/}\discretionary {}{}{}\texttt{gap\texttt{\symbol{45}}packages/}\discretionary {}{}{}\texttt{AutoDoc}}}, printedkey = {GH16} } @manual{ GAPDoc, author = {L{\"u}beck, F. and Neunh{\"o}ffer, M.}, title = {{GAPDoc (Version 1.6)}}, organization = {RWTH Aachen}, year = {2017}, note = {GAP package, \href {https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/index.html} {\texttt{https://www.math.rwth\texttt{\symbol{45}}aachen.de/}\discretionary {}{}{}\texttt{\texttt{\symbol{126}}Frank.Luebeck/}\discretionary {}{}{}\texttt{GAPDoc/}\discretionary {}{}{}\texttt{index.html}}}, printedkey = {LN17} } @manual{ GitHubPagesForGAP, author = {Horn, M.}, title = {{GitHubPagesForGAP \texttt{\symbol{45}} Template for easily using GitHub Pages within GAP packages (Version 0.2)}}, year = {2017}, note = {GAP package, \href {https://gap-system.github.io/GitHubPagesForGAP/} {\texttt{https://gap\texttt{\symbol{45}}system.github.io/}\discretionary {}{}{}\texttt{GitHubPagesForGAP/}}}, printedkey = {Hor17} } @manual{ MAGMA, author = {Bosma, W. and Cannon, J. and Playoust, C.}, title = {{The Magma algebra system. \texttt{\symbol{123}}I\texttt{\symbol{125}}. The user language}}, journal = {J. Symbolic Comput.}, volume = {24}, number = {3\texttt{\symbol{45}}4}, year = {1997}, pages = {235\texttt{\symbol{45}}\texttt{\symbol{45}}265}, note = {Computational algebra and number theory (London, 1993)\texttt{\symbol{125}} \href {https://doi.org/10.1006/jsco.1996.0125} {\texttt{https://doi.org/}\discretionary {}{}{}\texttt{10.1006/}\discretionary {}{}{}\texttt{jsco.1996.0125}}}, mrclass = {68Q40}, mrnumber = {MR1484478}, ISSN = {0747\texttt{\symbol{45}}7171}, doi = {10.1006/jsco.1996.0125}, printedkey = {BCP97} } @manual{ RCWA, author = {Kohl, S.}, title = {{RCWA \texttt{\symbol{45}} Residue\texttt{\symbol{45}}Class\texttt{\symbol{45}}Wise Affine Groups (Version 4.5.1)}}, year = {2017}, note = {GAP package, \href {https://stefan-kohl.github.io/rcwa.html} {\texttt{https://stefan\texttt{\symbol{45}}kohl.github.io/}\discretionary {}{}{}\texttt{rcwa.html}}}, printedkey = {Koh17} } @manual{ ResClasses, author = {Kohl, S.}, title = {{ResClasses \texttt{\symbol{45}} Set\texttt{\symbol{45}}Theoretic Computations with Residue Classes (Version 4.6.0)}}, year = {2017}, note = {GAP package, \href {https://stefan-kohl.github.io/resclasses.html} {\texttt{https://stefan\texttt{\symbol{45}}kohl.github.io/}\discretionary {}{}{}\texttt{resclasses.html}}}, printedkey = {Koh17} } @manual{ XMod, author = {Wensley, C. D. and Alp, M. and Odabas, A. and Uslu, E. O.}, title = {{XMod \texttt{\symbol{45}} Crossed Modules and Cat1\texttt{\symbol{45}}groups in GAP (Version 2.64)}}, year = {2017}, note = {GAP package, \href {https://github.com/gap-packages/xmod} {\texttt{https://github.com/}\discretionary {}{}{}\texttt{gap\texttt{\symbol{45}}packages/}\discretionary {}{}{}\texttt{xmod}}}, printedkey = {WAOU17} } utils-0.86/doc/chapInd.html000644 000765 000024 00000021222 14621610515 020714 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Index
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Index

AllIsomorphisms 7.1-1
AllIsomorphismsIterator 7.1-1
AllIsomorphismsNumber 7.1-1
AllProducts 4.1-2
AllSmoothIntegers 4.1-1 4.1-1
AllSubgroupsIterator 7.1-2
AssignGlobals 8.1-1
BlankFreeString 3.3-1
CartesianIterator 7.2-1
CentralProduct 5.3-3
CentralProductInfo 5.3-3
Comm 5.1-1
CommonRepresentatives 3.2-1
CommonTransversal 3.2-1
ConvertToMagmaInputString 10.3-1
CreateDirIfMissing 11.1
DifferencesList 3.1-1
DirectProductOfAutomorphismGroups 5.3-6
DirectProductOfFunctions 5.3-5
DirectSumDecompositionMatrices 6.1-1
distinct and common representatives 3.2
DistinctRepresentatives 3.2-1
Download 9.1-1
DownloadVerifyCertificate 9.1-2
EpimorphismByGenerators 5.3-1
ExponentOfPrime 11.3-2
FindMatchingFiles 11.1
Fitting series 5.1-5
FittingLength 5.1-5
FloatQuotientsList 3.1-2
GeneratorsAndInverses 5.1-4
GetSuffix 11.1
GitHub repository 1.
IdempotentEndomorphisms 5.3-4
IdempotentEndomorphismsData 5.3-4
IdempotentEndomorphismsWithImage 5.3-4
IntOrOnfinityToLaTeX 10.2-1
IsCommonTransversal 3.2-1
IsCommuting 5.1-2
Iterators 7.1
LaTeXStringFactorsInt 10.2-2
LeftCoset 5.2-1
ListOfPowers 5.1-3
Log2HTML 10.1-1
LowerFittingSeries 5.1-5
MatrixGroupToMagmaFormat 10.3-1
NextProbablyPrimeInt 4.1-4
OKtoReadFromUtils 12.
OptionRecordWithDefaults 8.2-1
PcGroupToMagmaFormat 10.3-1
PermGroupToMagmaFormat 10.3-1
PrimeNumbersIterator 4.1-5
PrintApplicableMethod 11.3-1
PrintOneItemPerLine 11.2
PrintSelection 2.1-1 2.1-1
Pullback 5.3-2
PullbackInfo 5.3-2
QuotientsList 3.1-2
RandomCombination 3.1-4
RestrictedPartitionsWithoutRepetitions 4.1-3
SearchCycle 3.1-3
SetIfMissing 11.1
smooth integer 4.1-1
StringDotSuffix 11.1
UnorderedPairsIterator 7.2-2
UpperFittingSeries 5.1-5

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3 Lists, Sets and Strings
 3.1 Functions for lists

  3.1-1 DifferencesList
  3.1-2 QuotientsList
  3.1-3 SearchCycle
  3.1-4 RandomCombination
 3.2 Distinct and Common Representatives

  3.2-1 DistinctRepresentatives
 3.3 Functions for strings

  3.3-1 BlankFreeString

3 Lists, Sets and Strings

3.1 Functions for lists

3.1-1 DifferencesList
‣ DifferencesList( L )( function )

This function has been transferred from package ResClasses.

It takes a list \(L\) of length \(n\) and outputs the list of length \(n-1\) containing all the differences \(L[i]-L[i-1]\).


gap> List( [1..12], n->n^3 );
[ 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 ]
gap> DifferencesList( last );
[ 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397 ]
gap> DifferencesList( last );
[ 12, 18, 24, 30, 36, 42, 48, 54, 60, 66 ]
gap> DifferencesList( last );
[ 6, 6, 6, 6, 6, 6, 6, 6, 6 ]

3.1-2 QuotientsList
‣ QuotientsList( L )( function )
‣ FloatQuotientsList( L )( function )

These functions have been transferred from package ResClasses.

They take a list \(L\) of length \(n\) and output the quotients \(L[i]/L[i-1]\) of consecutive entries in \(L\). An error is returned if an entry is zero.


gap> List( [0..10], n -> Factorial(n) );
[ 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 ]
gap> QuotientsList( last );
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]
gap> L := [ 1, 3, 5, -1, -3, -5 ];;
gap> QuotientsList( L );
[ 3, 5/3, -1/5, 3, 5/3 ]
gap> FloatQuotientsList( L );
[ 3., 1.66667, -0.2, 3., 1.66667 ]
gap> QuotientsList( [ 2, 1, 0, -1, -2 ] );
[ 1/2, 0, fail, 2 ]
gap> FloatQuotientsList( [1..10] );
[ 2., 1.5, 1.33333, 1.25, 1.2, 1.16667, 1.14286, 1.125, 1.11111 ]
gap> Product( last );
10.

3.1-3 SearchCycle
‣ SearchCycle( L )( operation )

This function has been transferred from package RCWA.

SearchCycle is a tool to find likely cycles in lists. What, precisely, a cycle is, is deliberately fuzzy here, and may possibly even change. The idea is that the beginning of the list may be anything, following that the same pattern needs to be repeated several times in order to be recognized as a cycle.


gap> L := [1..20];;  L[1]:=13;;                                              
gap> for i in [1..19] do                                                     
>        if IsOddInt(L[i]) then L[i+1]:=3*L[i]+1; else L[i+1]:=L[i]/2; fi;
>    od;                                                                  
gap> L;                                                                      
[ 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4 ]
gap> SearchCycle( L );                                                       
[ 1, 4, 2 ]
gap> n := 1;;  L := [n];;
gap> for i in [1..100] do  n:=(n^2+1) mod 1093;  Add(L,n);  od;
gap> L; 
[ 1, 2, 5, 26, 677, 363, 610, 481, 739, 715, 795, 272, 754, 157, 604, 848, 
  1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 
  211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 
  378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 
  754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 
  848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 
  271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004 ]
gap> C := SearchCycle( L );
[ 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754 ]
gap> P := Positions( L, 157 );
[ 14, 26, 38, 50, 62, 74, 86, 98 ]
gap> Length( C );  DifferencesList( P );
12
[ 12, 12, 12, 12, 12, 12, 12 ]

3.1-4 RandomCombination
‣ RandomCombination( S, k )( operation )

This function has been transferred from package ResClasses.

It returns a random unordered \(k\)-tuple of distinct elements of a set \(S\).


gap> ## "6 aus 49" is a common lottery in Germany
gap> RandomCombination( [1..49], 6 ); 
[ 2, 16, 24, 26, 37, 47 ]

3.2 Distinct and Common Representatives

3.2-1 DistinctRepresentatives
‣ DistinctRepresentatives( list )( operation )
‣ CommonRepresentatives( list )( operation )
‣ CommonTransversal( grp, subgrp )( operation )
‣ IsCommonTransversal( grp, subgrp, list )( operation )

These operations have been transferred from package XMod.

They deal with lists of subsets of \([1 \ldots n]\) and construct systems of distinct and common representatives using simple, non-recursive, combinatorial algorithms.

When \(L\) is a set of \(n\) subsets of \([1 \ldots n]\) and the Hall condition is satisfied (the union of any \(k\) subsets has at least \(k\) elements), a set of DistinctRepresentatives exists.

When \(J,K\) are both lists of \(n\) sets, the operation CommonRepresentatives returns two lists: the set of representatives, and a permutation of the subsets of the second list.

The operation CommonTransversal may be used to provide a common transversal for the sets of left and right cosets of a subgroup \(H\) of a group \(G\), although a greedy algorithm is usually quicker.


gap> J := [ [1,2,3], [3,4], [3,4], [1,2,4] ];;
gap> DistinctRepresentatives( J );
[ 1, 3, 4, 2 ]
gap> K := [ [3,4], [1,2], [2,3], [2,3,4] ];;
gap> CommonRepresentatives( J, K );
[ [ 3, 3, 3, 1 ], [ 1, 3, 4, 2 ] ]
gap> d16 := DihedralGroup( IsPermGroup, 16 ); 
Group([ (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) ])
gap> SetName( d16, "d16" );
gap> c4 := Subgroup( d16, [ d16.1^2 ] ); 
Group([ (1,3,5,7)(2,4,6,8) ])
gap> SetName( c4, "c4" );
gap> RightCosets( d16, c4 );
[ RightCoset(c4,()), RightCoset(c4,(2,8)(3,7)(4,6)), RightCoset(c4,(1,8,7,6,5,
   4,3,2)), RightCoset(c4,(1,8)(2,7)(3,6)(4,5)) ]
gap> trans := CommonTransversal( d16, c4 );
[ (), (2,8)(3,7)(4,6), (1,2,3,4,5,6,7,8), (1,2)(3,8)(4,7)(5,6) ]
gap> IsCommonTransversal( d16, c4, trans );
true

3.3 Functions for strings

3.3-1 BlankFreeString
‣ BlankFreeString( obj )( function )

This function has been transferred from package ResClasses.

The result of BlankFreeString( obj ); is a composite of the functions String( obj ) and RemoveCharacters( obj, " " );.


gap> gens := GeneratorsOfGroup( DihedralGroup(12) );
[ f1, f2, f3 ]
gap> String( gens );                                
"[ f1, f2, f3 ]"
gap> BlankFreeString( gens );                       
"[f1,f2,f3]"
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utils-0.86/doc/bib.xml000644 000765 000024 00000006312 14621444241 017742 0ustar00christopherwensleystaff000000 000000 SebastianGutsche MaxHorn <C>AutoDoc - Generate documentation from GAP source code (Version 2016.12.04)</C> 2016 GAP package, https://github.com/gap-packages/AutoDoc FrankLübeck MaxNeunhöffer <C>GAPDoc (Version 1.6)</C> RWTH Aachen 2017 GAP package, https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/index.html MaxHorn <C>GitHubPagesForGAP - Template for easily using GitHub Pages within GAP packages (Version 0.2)</C> 2017 GAP package, https://gap-system.github.io/GitHubPagesForGAP/ http://magma.maths.usyd.edu.au/magma/ WiebBosma JohnCannon CatherinePlayoust <C>The Magma algebra system. {I}. The user language</C> Computational algebra and number theory (London, 1993)} https://doi.org/10.1006/jsco.1996.0125 J. Symbolic Comput. 24 1997 3-4 235--265 0747-7171 68Q40 MR1484478 10.1006/jsco.1996.0125 StefanKohl <C>RCWA - Residue-Class-Wise Affine Groups (Version 4.5.1)</C> 2017 GAP package, https://stefan-kohl.github.io/rcwa.html StefanKohl <C>ResClasses - Set-Theoretic Computations with Residue Classes (Version 4.6.0)</C> 2017 GAP package, https://stefan-kohl.github.io/resclasses.html Chris D.Wensley MuratAlp AlperOdabas Enver OnderUslu <C>XMod - Crossed Modules and Cat1-groups in GAP (Version 2.64)</C> 2017 GAP package, https://github.com/gap-packages/xmod utils-0.86/doc/chapInd.txt000644 000765 000024 00000004623 14621610512 020572 0ustar00christopherwensleystaff000000 000000 Index AllIsomorphisms 7.1-1 AllIsomorphismsIterator 7.1-1 AllIsomorphismsNumber 7.1-1 AllProducts 4.1-2 AllSmoothIntegers 4.1-1 4.1-1 AllSubgroupsIterator 7.1-2 AssignGlobals 8.1-1 BlankFreeString 3.3-1 CartesianIterator 7.2-1 CentralProduct 5.3-3 CentralProductInfo 5.3-3 Comm 5.1-1 CommonRepresentatives 3.2-1 CommonTransversal 3.2-1 ConvertToMagmaInputString 10.3-1 CreateDirIfMissing 11.1 DifferencesList 3.1-1 DirectProductOfAutomorphismGroups 5.3-6 DirectProductOfFunctions 5.3-5 DirectSumDecompositionMatrices 6.1-1 distinct and common representatives 3.2 DistinctRepresentatives 3.2-1 Download 9.1-1 DownloadVerifyCertificate 9.1-2 EpimorphismByGenerators 5.3-1 ExponentOfPrime 11.3-2 FindMatchingFiles 11.1 Fitting series 5.1-5 FittingLength 5.1-5 FloatQuotientsList 3.1-2 GeneratorsAndInverses 5.1-4 GetSuffix 11.1 GitHub repository 1.0 IdempotentEndomorphisms 5.3-4 IdempotentEndomorphismsData 5.3-4 IdempotentEndomorphismsWithImage 5.3-4 IntOrOnfinityToLaTeX 10.2-1 IsCommonTransversal 3.2-1 IsCommuting 5.1-2 Iterators 7.1 LaTeXStringFactorsInt 10.2-2 LeftCoset 5.2-1 ListOfPowers 5.1-3 Log2HTML 10.1-1 LowerFittingSeries 5.1-5 MatrixGroupToMagmaFormat 10.3-1 NextProbablyPrimeInt 4.1-4 OKtoReadFromUtils 12.0 OptionRecordWithDefaults 8.2-1 PcGroupToMagmaFormat 10.3-1 PermGroupToMagmaFormat 10.3-1 PrimeNumbersIterator 4.1-5 PrintApplicableMethod 11.3-1 PrintOneItemPerLine 11.2 PrintSelection 2.1-1 2.1-1 Pullback 5.3-2 PullbackInfo 5.3-2 QuotientsList 3.1-2 RandomCombination 3.1-4 RestrictedPartitionsWithoutRepetitions 4.1-3 SearchCycle 3.1-3 SetIfMissing 11.1 smooth integer 4.1-1 StringDotSuffix 11.1 UnorderedPairsIterator 7.2-2 UpperFittingSeries 5.1-5 ------------------------------------------------------- utils-0.86/doc/chap12.txt000644 000765 000024 00000013105 14621610512 020275 0ustar00christopherwensleystaff000000 000000 12 The transfer procedure We consider here the process for transferring utility functions from a package Home to Utils which has to avoid the potential problem of duplicate declarations of a function causing loading problems in GAP. If the functions in Home all have names of the form HOME_FunctionName then, in Utils, these functions are likely to be renamed as FunctionName or something similar. In this case the problem of duplicate declarations does not arise. This is what has happened with transfers from the AutoDoc package. The case where the function names are unchanged is more complicated. Initially we tried out a process which allowed repeated declarations and installations of the functions being transferred. This involved additions to the main library files global.g and oper.g. Since there were misgivings about interfering in this way with basic operations such as BIND_GLOBAL, a simpler (but slightly less convenient) process has been adopted. Using this alternative procedure, the following steps will be followed when making transfers from Home to Utils. 1 (Home:) Offer functions for inclusion. This may be simply done by emailing a list of functions. More usefully, email the declaration, implementation, test and documentation files, e.g.: home.gd, home.gi, home.tst and home.xml. (All active authors should be involved.) 2 (Home:) Declare that m.n is the last version of Home to contain these functions, so that m.n+1 (or similar) will be the first version of Home to have all these functions removed, and to specify Utils as a required package. 3 (Utils:) Add strings "home" and "m.n" to the list UtilsPackageVersions in the file utils/lib/start.gd.   Example   UtilsPackageVersions :=   [ "autodoc", "2016.01.31",   "resclasses", "4.2.5",   "home", "m.n",  ..., ...   ];   While the transfers are being made, it is essential that any new versions of Home should be tested with the latest version of Utils before they are released, so as to avoid loading failures. 4 (Utils:) Include the function declaration and implementation sections in suitable files, enclosed within a conditional clause of the form:   Example   if OKtoReadFromUtils( "Home" ) then . . . . . .    . . . . . .  fi;   The function OKtoReadFromUtils returns true only if there is an installed version of Home and if this version is greater than m.n. So, at this stage, the copied code will not be read, and the transferred functions can only be called if Home has been installed. 5 (Utils:) Add the test and documentation material to the appropriate files. The copied code can be tested by temporarily moving Home away from GAP's package directory. 6 (Utils:) Release a new version of Utils containing all the transferred material. 7 (Home:) Edit out the declarations and implementations of all the transferred functions, and remove references to them in the manual and tests. Possibly add a note to the manual that these functions have been transferred. Add Utils to the list of Home's required packages in PackageInfo.g. Release a new version of Home. 8 (Utils:) In due course, when the new version(s) of Home are well established, it may be safe to remove the conditional clauses mentioned in item 4 above. The entry for Home in UtilsPackageLists may then be removed. Finally, a note on the procedure for testing these functions. As long as a function being transferred still exists in the Home package, the code will not be read from Utils. So, when the tests are run, it is necessary to LoadPackage("home") before the function is called. The file utils/tst/testall.g makes sure that all the necessary packages are loaded before the individual tests are called. utils-0.86/doc/chap4.html000644 000765 000024 00000030177 14621610515 020356 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Chapter 4: Number-theoretic functions
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4 Number-theoretic functions
 4.1 Functions for integers

  4.1-1 AllSmoothIntegers
  4.1-2 AllProducts
  4.1-3 RestrictedPartitionsWithoutRepetitions
  4.1-4 NextProbablyPrimeInt
  4.1-5 PrimeNumbersIterator

4 Number-theoretic functions

4.1 Functions for integers

4.1-1 AllSmoothIntegers
‣ AllSmoothIntegers( maxp, maxn )( function )
‣ AllSmoothIntegers( L, maxp )( function )

This function has been transferred from package RCWA.

The function AllSmoothIntegers(maxp,maxn) returns the list of all positive integers less than or equal to maxn whose prime factors are all in the list L = {p ~|~ p leqslant maxp, p~mboxprime }.

In the alternative form, when L is a list of primes, the function returns the list of all positive integers whose prime factors lie in L.


gap> AllSmoothIntegers( 3, 1000 );
[ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 
  108, 128, 144, 162, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576, 
  648, 729, 768, 864, 972 ]
gap> AllSmoothIntegers( [5,11,17], 1000 );
[ 1, 5, 11, 17, 25, 55, 85, 121, 125, 187, 275, 289, 425, 605, 625, 935 ]
gap> Length( last );
16
gap> List( [3..20], n -> Length( AllSmoothIntegers( [5,11,17], 10^n ) ) );
[ 16, 29, 50, 78, 114, 155, 212, 282, 359, 452, 565, 691, 831, 992, 1173, 
  1374, 1595, 1843 ]

4.1-2 AllProducts
‣ AllProducts( L, k )( function )

This function has been transferred from package RCWA.

The command AllProducts(L,k) returns the list of all products of k entries of the list L. Note that every ordering of the entries is used so that, in the commuting case, there are bound to be repetitions.


gap> AllProducts([1..4],3); 
[ 1, 2, 3, 4, 2, 4, 6, 8, 3, 6, 9, 12, 4, 8, 12, 16, 2, 4, 6, 8, 4, 8, 12, 
  16, 6, 12, 18, 24, 8, 16, 24, 32, 3, 6, 9, 12, 6, 12, 18, 24, 9, 18, 27, 
  36, 12, 24, 36, 48, 4, 8, 12, 16, 8, 16, 24, 32, 12, 24, 36, 48, 16, 32, 
  48, 64 ]
gap> Set(last);            
[ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 64 ]
gap> AllProducts( [(1,2,3),(2,3,4)], 2 );
[ (2,4,3), (1,2)(3,4), (1,3)(2,4), (1,3,2) ]

4.1-3 RestrictedPartitionsWithoutRepetitions
‣ RestrictedPartitionsWithoutRepetitions( n, S )( function )

This function has been transferred from package RCWA.

For a positive integer n and a set of positive integers S, this function returns the list of partitions of n into distinct elements of S. Unlike RestrictedPartitions, no repetitions are allowed.


gap> RestrictedPartitions( 20, [4..10] );
[ [ 4, 4, 4, 4, 4 ], [ 5, 5, 5, 5 ], [ 6, 5, 5, 4 ], [ 6, 6, 4, 4 ], 
  [ 7, 5, 4, 4 ], [ 7, 7, 6 ], [ 8, 4, 4, 4 ], [ 8, 6, 6 ], [ 8, 7, 5 ], 
  [ 8, 8, 4 ], [ 9, 6, 5 ], [ 9, 7, 4 ], [ 10, 5, 5 ], [ 10, 6, 4 ], 
  [ 10, 10 ] ]
gap> RestrictedPartitionsWithoutRepetitions( 20, [4..10] );
[ [ 10, 6, 4 ], [ 9, 7, 4 ], [ 9, 6, 5 ], [ 8, 7, 5 ] ]
gap> RestrictedPartitionsWithoutRepetitions( 10^2, List([1..10], n->n^2 ) );
[ [ 100 ], [ 64, 36 ], [ 49, 25, 16, 9, 1 ] ]

4.1-4 NextProbablyPrimeInt
‣ NextProbablyPrimeInt( n )( function )

This function has been transferred from package RCWA.

The function NextProbablyPrimeInt(n) does the same as NextPrimeInt(n) except that for reasons of performance it tests numbers only for IsProbablyPrimeInt(n) instead of IsPrimeInt(n). For large n, this function is much faster than NextPrimeInt(n)


gap> n := 2^251;
3618502788666131106986593281521497120414687020801267626233049500247285301248
gap> NextProbablyPrimeInt( n );
3618502788666131106986593281521497120414687020801267626233049500247285301313
gap> time;                     
1
gap> NextPrimeInt( n );        
3618502788666131106986593281521497120414687020801267626233049500247285301313
gap> time;             
213

4.1-5 PrimeNumbersIterator
‣ PrimeNumbersIterator( [chunksize] )( function )

This function has been transferred from package RCWA.

This function returns an iterator which runs over the prime numbers n ascending order; it takes an optional argument chunksize which specifies the length of the interval which is sieved in one go (the default is 10^7), and which can be used to balance runtime vs. memory consumption. It is assumed that chunksize is larger than any gap between two consecutive primes within the range one intends to run the iterator over.


gap> iter := PrimeNumbersIterator();;
gap> for i in [1..100] do  p := NextIterator(iter);  od;
gap> p;
541
gap> sum := 0;;
gap> ## "prime number race" 1 vs. 3 mod 4
gap> for p in PrimeNumbersIterator() do 
>       if p <> 2 then sum := sum + E(4)^(p-1); fi;
>       if sum > 0 then break; fi;
>    od;
gap> p;
26861
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utils-0.86/doc/others.xml000644 000765 000024 00000011413 14621444241 020510 0ustar00christopherwensleystaff000000 000000 Various other functions
File operations This function has been transferred from package &RCWA;.

This function converts the &GAP; logfile filename to HTML. It appears that the logfile should be in your current directory. The extension of the input file must be *.log. The name of the output file is the same as the one of the input file except that the extension *.log is replaced by *.html. There is a sample CSS file in utils/doc/gaplog.css, which you can adjust to your taste.

LogTo( "triv.log" ); gap> a := 33^5; 39135393 gap> LogTo(); gap> Log2HTML( "triv.log" ); ]]>

&LaTeX; strings This function has been transferred from package &ResClasses;.

IntOrInfinityToLaTeX(n) returns the &LaTeX; string for n.

IntOrInfinityToLaTeX( 10^3 ); "1000" gap> IntOrInfinityToLaTeX( infinity ); "\\infty" ]]> This function has been transferred from package &RCWA;.

It returns the prime factorization of the integer n as a string in &LaTeX; format.

LaTeXStringFactorsInt( Factorial(12) ); "2^{10} \\cdot 3^5 \\cdot 5^2 \\cdot 7 \\cdot 11" ]]>

Conversion to &Magma; string The function ConvertToMagmaInputString( obj [, str] ) attempts to output a string s which can be read into &Magma; so as to produce the same group in that computer algebra system. In the second form the user specifies the name of the resulting object, so that the output string has the form "str := ...". When obj is a permutation group, the operation PermGroupToMagmaFormat(obj) is called. PermGroupToMagmaFormat This function has been taken from other.gi in the main library where it was called MagmaInputString. When obj is a pc-group, the operation PcGroupToMagmaFormat(obj) is called. PcGroupToMagmaFormat This function was private code of Max Horn. When obj is a matrix group over a finite field, the operation MatrixGroupToMagmaFormat(obj) is called. MatrixGroupToMagmaFormat This function is a modification of private code of Frank Lübeck.

Hopefully code for other types of group will be added in due course.

These functions should be considered experimental, and more testing is desirable.

ConvertToMagmaInputString( Group( (1,2,3,4,5), (3,4,5) ) ); "PermutationGroup<5|(1,2,3,4,5),\n(3,4,5)>;\n" gap> ConvertToMagmaInputString( Group( (1,2,3,4,5) ), "c5" ); "c5:=PermutationGroup<5|(1,2,3,4,5)>;\n" gap> ConvertToMagmaInputString( DihedralGroup( IsPcGroup, 10 ) ); "PolycyclicGroup< f1,f2 |\nf1^2,\nf2^5,\nf2^f1 = f2^4\n>;\n" gap> M := GL(2,5);; Size(M); 480 gap> s1 := ConvertToMagmaInputString( M ); "F := GF(5);\nP := GL(2,F);\ngens := [\nP![2,0,0,1],\nP![4,1,4,0]\n];\nsub

;\n" gap> Print( s1 ); F := GF(5); P := GL(2,F); gens := [ P![2,0,0,1], P![4,1,4,0] ]; sub

; gap> n1 := [ [ Z(9)^0, Z(9)^0 ], [ Z(9)^0, Z(9) ] ];; gap> n2 := [ [ Z(9)^0, Z(9)^3 ], [ Z(9)^4, Z(9)^2 ] ];; gap> N := Group( n1, n2 );; Size( N ); 5760 gap> s2 := ConvertToMagmaInputString( N, "gpN" );; gap> Print( s2 ); F := GF(3^2); P := GL(2,F); w := PrimitiveElement(F); gens := [ P![ 1, 1, 1,w^1], P![ 1,w^3, 2,w^2] ]; gpN := sub

; ]]>

utils-0.86/doc/chap10.txt000644 000765 000024 00000013466 14621610512 020305 0ustar00christopherwensleystaff000000 000000 10 Various other functions 10.1 File operations 10.1-1 Log2HTML Log2HTML( filename )  function This function has been transferred from package RCWA. This function converts the GAP logfile filename to HTML. It appears that the logfile should be in your current directory. The extension of the input file must be *.log. The name of the output file is the same as the one of the input file except that the extension *.log is replaced by *.html. There is a sample CSS file in utils/doc/gaplog.css, which you can adjust to your taste.  Example   gap> LogTo( "triv.log" ); gap> a := 33^5; 39135393 gap> LogTo();  gap> Log2HTML( "triv.log" );    10.2 LaTeX strings 10.2-1 IntOrOnfinityToLaTeX IntOrOnfinityToLaTeX( n )  function This function has been transferred from package ResClasses. IntOrInfinityToLaTeX(n) returns the LaTeX string for n.  Example   gap> IntOrInfinityToLaTeX( 10^3 ); "1000" gap> IntOrInfinityToLaTeX( infinity ); "\\infty"   10.2-2 LaTeXStringFactorsInt LaTeXStringFactorsInt( n )  function This function has been transferred from package RCWA. It returns the prime factorization of the integer n as a string in LaTeX format.  Example   gap> LaTeXStringFactorsInt( Factorial(12) ); "2^{10} \\cdot 3^5 \\cdot 5^2 \\cdot 7 \\cdot 11"   10.3 Conversion to Magma string 10.3-1 ConvertToMagmaInputString ConvertToMagmaInputString( arg )  function The function ConvertToMagmaInputString( obj [, str] ) attempts to output a string s which can be read into Magma [BCP97] so as to produce the same group in that computer algebra system. In the second form the user specifies the name of the resulting object, so that the output string has the form "str := ...". When obj is a permutation group, the operation PermGroupToMagmaFormat(obj) is called. This function has been taken from other.gi in the main library where it was called MagmaInputString. When obj is a pc-group, the operation PcGroupToMagmaFormat(obj) is called. This function was private code of Max Horn. When obj is a matrix group over a finite field, the operation MatrixGroupToMagmaFormat(obj) is called. This function is a modification of private code of Frank Lübeck. Hopefully code for other types of group will be added in due course. These functions should be considered experimental, and more testing is desirable.  Example   gap> ConvertToMagmaInputString( Group( (1,2,3,4,5), (3,4,5) ) ); "PermutationGroup<5|(1,2,3,4,5),\n(3,4,5)>;\n" gap> ConvertToMagmaInputString( Group( (1,2,3,4,5) ), "c5" );  "c5:=PermutationGroup<5|(1,2,3,4,5)>;\n" gap> ConvertToMagmaInputString( DihedralGroup( IsPcGroup, 10 ) ); "PolycyclicGroup< f1,f2 |\nf1^2,\nf2^5,\nf2^f1 = f2^4\n>;\n" gap> M := GL(2,5);; Size(M);  480 gap> s1 := ConvertToMagmaInputString( M ); "F := GF(5);\nP := GL(2,F);\ngens := [\nP![2,0,0,1],\nP![4,1,4,0]\n];\nsub

;\n" gap> Print( s1 ); F := GF(5); P := GL(2,F); gens := [ P![2,0,0,1], P![4,1,4,0] ]; sub

; gap> n1 := [ [ Z(9)^0, Z(9)^0 ], [ Z(9)^0, Z(9) ] ];; gap> n2 := [ [ Z(9)^0, Z(9)^3 ], [ Z(9)^4, Z(9)^2 ] ];; gap> N := Group( n1, n2 );; Size( N ); 5760 gap> s2 := ConvertToMagmaInputString( N, "gpN" );; gap> Print( s2 ); F := GF(3^2); P := GL(2,F); w := PrimitiveElement(F); gens := [ P![ 1, 1, 1,w^1], P![ 1,w^3, 2,w^2] ]; gpN := sub

;   utils-0.86/doc/chap11.txt000644 000765 000024 00000010414 14621610512 020274 0ustar00christopherwensleystaff000000 000000 11 Obsolete functions 11.1 Operations from AutoDoc The file functions FindMatchingFiles and CreateDirIfMissing were copied from package AutoDoc where they are named AutoDoc_FindMatchingFiles and AutoDoc_CreateDirIfMissing. The string function StringDotSuffix was also copied from package AutoDoc, where it is named AUTODOC_GetSuffix. The function SetIfMissing was also transferred from package AutoDoc, where it is called AUTODOC_SetIfMissing. It writes into a record provided the position is not yet bound. As from version 0.61, all these functions became obsolete in Utils, but continue to be defined in AutoDoc. 11.2 Functions for printing The function PrintOneItemPerLine was used to prints lists vertically, rather than horizontally. Since a very similar result may be achieved using the GAP library functions Perform and Display, this function became obsolete in version 0.61.  Example   gap> s3 := SymmetricGroup( 3 );;  gap> L := KnownPropertiesOfObject( GeneratorsOfGroup( s3 ) );; gap> Perform( L, Display ); IsFinite IsSmallList IsGeneratorsOfMagmaWithInverses IsGeneratorsOfSemigroup IsSubsetLocallyFiniteGroup gap> Perform( s3, Display );  () (2,3) (1,3) (1,3,2) (1,2,3) (1,2)   11.3 Other obsolete functions 11.3-1 Applicable Methods The function PrintApplicableMethod, which was included in versions from 0.41 to 0.58, has been removed since it was considered superfluous. The example shows how to print out a function.  Example   gap> ApplicableMethod( IsCyclic, [ Group((1,2,3),(4,5)) ], 1, 1 ); #I Searching Method for IsCyclic with 1 arguments: #I Total: 7 entries #I Method 4: ``IsCyclic'' at /Applications/gap/gap4r9/lib/grp.gi:30 , value:  36 function( G ) ... end gap> Print( last ); function ( G )  if Length( GeneratorsOfGroup( G ) ) = 1 then  return true;  else  TryNextMethod();  fi;  return; end gap> ApplicableMethod( IsCyclic, [ Group((1,2,3),(4,5)) ], 0, 3 ); function( <1 unnamed arguments> ) ... end gap> Print( last );  function ( <> )  <> end   11.3-2 ExponentOfPrime The function ExponentOfPrime was originally transferred from package RCWA. The command ExponentOfPrime(n,p) returned the exponent of the prime p in the prime factorization of n. Since the GAP function PValuation produces the same results, and does so more quickly, this function has been made obsolete. utils-0.86/doc/chap8.html000644 000765 000024 00000023004 14621610515 020351 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Chapter 8: Records

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8 Records
 8.1 Functions for records

  8.1-1 AssignGlobals
 8.2 Option records for functions

  8.2-1 OptionRecordWithDefaults

8 Records

8.1 Functions for records

8.1-1 AssignGlobals
‣ AssignGlobals( rec )( function )

This function has been transferred from package RCWA.

It assigns the record components of rec to global variables with the same names.


gap> r := rec( a := 1, b := 2, c := 3 );;                                      
gap> AssignGlobals( r );
The following global variables have been assigned:
[ "a", "b", "c" ]
gap> [a,b,c];
[ 1, 2, 3 ]

8.2 Option records for functions

8.2-1 OptionRecordWithDefaults
‣ OptionRecordWithDefaults( defaults, useroptions )( function )

This functions has been transferred by Chris Jefferson from other packages. It simplifies the handling of records which are intended to be used for expressing configuration options. defaults represents the "default record", and useroptions lets the user give new values for values in defaults.

The function returns a record with the same component names as defaults and which has the same values as defaults, except for those component names in useroptions, where the values in useroptions are used instead. An error is given if useroptions contains any component names not in defaults. If useroptions is an empty list it is treated as an empty record, and if useroptions is a list of length 1 containing a record, this record is used as useroptions.


gap> defaults := rec( a := 1, b := 2, c := 3 );;
gap> OptionRecordWithDefaults( defaults, rec( a := 6) );
rec( a := 6, b := 2, c := 3 )
gap> OptionRecordWithDefaults( defaults, rec( b := 7, c := 8 ) );
rec( a := 1, b := 7, c := 8 )
gap> OptionRecordWithDefaults( defaults, [ ] );
rec( a := 1, b := 2, c := 3 )
gap> OptionRecordWithDefaults( defaults, [ rec( c := 8 ) ] );
rec( a := 1, b := 2, c := 8 )
gap> OptionRecordWithDefaults( defaults, rec( d := 9 ) );
Error, Unknown option: d
gap> OptionRecordWithDefaults( defaults, [ rec( b := 7 ), rec( c := 8 ) ] );
Error, Too many arguments for function
gap> OptionRecordWithDefaults( defaults, [6,7,8] );
Error, Too many arguments for function

This function is designed to support functions with optional arguments given as a variable record, of the form function(x,y,options...). In the following, very contrived, example function, PrintDimensions, the defaults are given by the variable order which takes values h, w and d having default values 1, 2 and 3. If there is a second argument, then OptionRecordWithDefaults( order, arg[2] ); is used to cvhange the values. These three values then determine the order in which the three dimensions are printed using a SortParallel command.


PrintDimensions := function( arg ) 
    local nargs, dim, order, V, L, len, K, i; 
    nargs := Length( arg ); 
    dim := [ arg[1]!.height, arg[1]!.width, arg[1]!.depth ]; 
    order := rec( h := 1, w := 2, d := 3 ); 
    V := [ "height", "width", "depth" ]; 
    if ( nargs > 1 ) and IsRecord( arg[2] ) then 
        order := OptionRecordWithDefaults( order, arg[2] ); 
    fi; 
    L := [ order!.h, order!.w, order!.d ]; 
    len := Length( L );
    K := [ 1..len ]; 
    SortParallel( L, K ); 
    Print( "dimensions: " ); 
    Print( V[K[1]], " = ", dim[K[1]], ", " );
    Print( V[K[2]], " = ", dim[K[2]], ", " );
    Print( V[K[3]], " = ", dim[K[3]], "\n" );
end;;

In the example below the first call to PrintDimensions has just one parameter, mydim, so the default order is used. In the second call, alternate values for h, w and d are given, causing the width to be printed first, and then the depth and height.


gap> mydim := rec( height := 45, width := 31, depth := 17 ); 
rec( depth := 17, height := 45, width := 31 )
gap> PrintDimensions( mydim );
dimensions: height = 45, width = 31, depth = 17
gap> PrintDimensions( mydim, rec( h:=3, w:=1, d:=2 ) );
dimensions: width = 31, depth = 17, height = 45
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9 Web Downloads
 9.1 Functions for downloading files from the web

  9.1-1 Download
  9.1-2 User preference DownloadVerifyCertificate

9 Web Downloads

The Download operation has been written by Thomas Breuer, incorporating a number of suggestions from Max Horn, for version 0.77 of Utils. It implements downloading a file from within GAP. It can use the IO or curlInterface packages, or wget or curl, if installed, and it can be extended with other download methods quite easily. It is envisaged that, once other packages have started to use it, and any problems have been addressed, that the functions will be transferred to the main GAP library.

9.1 Functions for downloading files from the web

9.1-1 Download
‣ Download( url[, opt] )( function )

This function downloads the file with the web address url, which must be a string.

The result is a record which has at least the component success, with value true if the download was successful and false otherwise. In the former case, the component result is bound, whose value is a string that contains the contents of the downloaded file. In the latter case, the component error is bound, whose value is a string that describes the problem.

The function calls the methods stored in the global list Download_Methods until one of them is successful. Currently there are methods based on the GAP functions DownloadURL (curl: DownloadURL) and SingleHTTPRequest (IO: SingleHTTPRequest), and methods based on the external programs wget and curl.

An optional record opt can be given. The following components are supported.

target

If this component is bound then its value must be a string that is a local filename, and the function writes the downloaded contents to this file; the returned record does not have a result component in this case.

verifyCert

If this component is bound and has the value false then those download methods that are based on curl or wget will omit the check of the server's certificate. The same effect is achieved for all Download calls by setting the user preference DownloadVerifyCertificate (see 9.1-2) to false and omitting the verifyCert component from opt.


gap> url:= "https://www.gap-system.org/Packages/utils.html";;
gap> res1:= Download( url );;
gap> res1.success;
true
gap> IsBound( res1.result ) and IsString( res1.result );
true
gap> res2:= Download( Concatenation( url, "xxx" ) );;
gap> res2.success;
false
gap> IsBound( res2.error ) and IsString( res2.error );
true

9.1-2 User preference DownloadVerifyCertificate

The value true (the default) means that the server's certificate is checked in calls of Download (9.1-1), such that nothing gets downloaded if the certificate is invalid.

If the value is false then download methods are supposed to omit the check of the server's certificate (this may not be supported by all download methods).

One can set the value of the preference to be val via SetUserPreference (Reference: SetUserPreference), by calling SetUserPreference( "utils", "DownloadVerifyCertificate", val ), and access the current value via UserPreference (Reference: UserPreference), by calling UserPreference( "utils", "DownloadVerifyCertificate" ).

We recommend leaving this preference at its default value true. Sometimes it can be necessary to change it, e.g. to work around issues with old operating systems which may not be able to correctly verify new certificates. In general it is better to update such a system, but if that is not an option, then disabling certificate checks may be a good last resort.

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utils-0.86/doc/intro.xml000644 000765 000024 00000011663 14621444241 020346 0ustar00christopherwensleystaff000000 000000 Introduction The &Utils; package provides a space for utility functions from a variety of &GAP; packages to be collected together into a single package. In this way it is hoped that they will become more visible to other package authors. This package was first distributed as part of the &GAP; 4.8.2 distribution.

The package is loaded with the command LoadPackage( "utils" ); ]]>

Functions have been transferred from the following packages: Conversion of a &GAP; group to a &Magma; output string, taken from various sources including other.gi in the main library.

Transfer is complete (for now) for functions from the following packages: &AutoDoc; (with function names changed); &ResClasses; ; &RCWA; ; &XMod; .

The package may be obtained either as a compressed .tar file or as a .zip file, utils-version_number.tar.gz, by ftp from one of the following sites: the &Utils; GitHub release site: https://gap-packages.github.io/utils/. any &GAP; archive, e.g. https://www.gap-system.org/Packages/packages.html; GitHub repository The package also has a GitHub repository at: https://github.com/gap-packages/utils.

Once the package is loaded, the manual doc/manual.pdf can be found in the documentation folder. The html versions, with or without &MathJax;, may be rebuilt as follows:

ReadPackage( "utils", "makedoc.g" ); ]]>

It is possible to check that the package has been installed correctly by running the test files (which terminates the &GAP; session):

ReadPackage( "utils", "tst/testall.g" ); Architecture: . . . . . testing: . . . . . . . . #I No errors detected while testing ]]>

Note that functions listed in this manual that are currently in the process of being transferred are only read from the source package &Home; (say), and so can only be used if &Home; has already been loaded. There are no such functions in transition at present.

Information for package authors A function (or collection of functions) is suitable for transfer from a package &Home; to &Utils; if the following conditions are satisfied. The function is sufficiently non-specialised so that it might be of use to other authors. The function does not depend on the remaining functions in &Home; The function does not do what can already be done with a &GAP; library function. Documentation of the function and test examples are available. When there is more than one active author of &Home;, they should all be aware (and content) that the transfer is taking place.

Authors of packages may be reluctant to let go of their utility functions. The following principles may help to reassure them. (Suggestions for more items here are welcome.) A function that has been transferred to &Utils; will not be changed without the approval of the original author. The current package maintainer has every intention of continuing to maintain &Utils;. In the event that this proves impossible, the &GAP; development team will surely find someone to take over. Function names will not be changed unless specifically requested by &Home;'s author(s) or unless they have the form HOME_FunctionName. In order to speed up the transfer process, only functions from one package will be in transition at any given time. Hopefully a week or two will suffice for most packages. Any package author who transfers a function to &Utils; will become an author of &Utils;. (In truth, &Utils; does not have authors, just a large number of contributors.)

The process for transferring utility functions from &Home; to &Utils; is described in Chapter .

utils-0.86/doc/matrix.xml000644 000765 000024 00000007731 14621444241 020520 0ustar00christopherwensleystaff000000 000000 Matrices
Some operations for matrices In June 2023 Hongyi Zhao asked in the Forum for a function to implement matrix decomposition into blocks. Such a function was then provided by Pedro García-Sánchez. Hongyi Zhao then requested that the function be added to Utils. What is provided here is a revised version of the original solution, returning a list of decompositions.

This function is a partial inverse to the undocumented library operation DirectSumMat. So if L is the list of diagonal decompositions of a matrix M then each entry in L is a list of matrices, and the direct sum of each of these lists is equal to the original M.

In the following examples, M_6 is an obvious direct sum with 3 blocks. M_4 is an example with three decompositions, while M_8 = M_4 \oplus M_4 has 16 decompositions (not listed).

M6 := [ [1,2,0,0,0,0], [3,4,0,0,0,0], [5,6,0,0,0,0], > [0,0,9,0,0,0], [0,0,0,1,2,3], [0,0,0,4,5,6] ];; gap> Display( M6 ); [ [ 1, 2, 0, 0, 0, 0 ], [ 3, 4, 0, 0, 0, 0 ], [ 5, 6, 0, 0, 0, 0 ], [ 0, 0, 9, 0, 0, 0 ], [ 0, 0, 0, 1, 2, 3 ], [ 0, 0, 0, 4, 5, 6 ] ] gap> L6 := DirectSumDecompositionMatrices( M6 ); [ [ [ [ 1, 2 ], [ 3, 4 ], [ 5, 6 ] ], [ [ 9 ] ], [ [ 1, 2, 3 ], [ 4, 5, 6 ] ] ] ] gap> M4 := [ [0,3,0,0], [0,0,0,0], [0,0,0,0], [0,0,4,0] ];; gap> Display( M4 ); [ [ 0, 3, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 4, 0 ] ] gap> L4 := DirectSumDecompositionMatrices( M4 ); [ [ [ [ 0, 3 ] ], [ [ 0, 0 ], [ 0, 0 ], [ 4, 0 ] ] ], [ [ [ 0, 3 ], [ 0, 0 ] ], [ [ 0, 0 ], [ 4, 0 ] ] ], [ [ [ 0, 3 ], [ 0, 0 ], [ 0, 0 ] ], [ [ 4, 0 ] ] ] ] gap> for L in L4 do > A := DirectSumMat( L );; > if ( A = M4 ) then Print( "yes, A = M4\n" ); fi; > od; yes, A = M4 yes, A = M4 yes, A = M4 gap> M8 := DirectSumMat( M4, M4 );; gap> Display( M8 ); [ [ 0, 3, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 4, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 3, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 4, 0 ] ] gap> L8 := DirectSumDecompositionMatrices( M8 );; gap> Length( L8 ); 16 ]]>

The current method does not, however, catch all possible decompositions. In the following example the matrix M_5 has its third row and third column extirely zero, and the only decomposition found has a [0] factor. There are clearly two 2-factor decompositions with a 2-by-3 and a 3-by-2 factor, but these are not found at present.

M5 := [ [1,2,0,0,0], [3,4,0,0,0], [0,0,0,0,0], > [0,0,0,6,7], [0,0,0,8,9] ];; gap> Display(M5); [ [ 1, 2, 0, 0, 0 ], [ 3, 4, 0, 0, 0 ], [ 0, 0, 0, 0, 0 ], [ 0, 0, 0, 6, 7 ], [ 0, 0, 0, 8, 9 ] ] gap> L5 := DirectSumDecompositionMatrices( M5 ); [ [ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 0 ] ], [ [ 6, 7 ], [ 8, 9 ] ] ] ] ]]>

utils-0.86/doc/chap2_mj.html000644 000765 000024 00000013630 14621610515 021035 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Chapter 2: Printing Lists and Iterators
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2 Printing Lists and Iterators
 2.1 Printing selected items

  2.1-1 PrintSelection

2 Printing Lists and Iterators

2.1 Printing selected items

The functions described here print lists or objects with an iterator with one item per line, either the whole list/iterator or certain subsets:

  • by giving a list of positions of items to be printed, or

  • by specifying a first item and then a regular step.

2.1-1 PrintSelection
‣ PrintSelection( obj, first, step[, last] )( function )
‣ PrintSelection( obj, list )( function )

This function, given three (or four) parameters, calls operations PrintSelectionFromList or PrintSelectionFromIterator which prints the first item specified, and then the item at every step. The fourth parameter is essential when the object being printed is infinite.

Alternatively, given two parameters, with the second parameter a list L of positive integers, only the items at positions in L are printed.


gap> L := List( [1..20], n -> n^5 );;
gap> PrintSelection( L, [18..20] );
18 : 1889568
19 : 2476099
20 : 3200000
gap> PrintSelection( L, 2, 9 );       
2 : 32
11 : 161051
20 : 3200000
gap> PrintSelection( L, 2, 3, 11 );
2 : 32
5 : 3125
8 : 32768
11 : 161051
gap> s5 := SymmetricGroup( 5 );;
gap> PrintSelection( s5, [30,31,100,101] );
30 : (1,5)(3,4)
31 : (1,5,2)
100 : (1,4,3)
101 : (1,4)(3,5)
gap> PrintSelection( s5, 1, 30 );
1 : ()
31 : (1,5,2)
61 : (1,2,3)
91 : (1,3,5,2,4)
gap> PrintSelection( s5, 9, 11, 43 );
9 : (2,5,3)
20 : (2,4)
31 : (1,5,2)
42 : (1,5,2,3,4)
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utils-0.86/doc/chap5.html000644 000765 000024 00000103715 14621610515 020356 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Chapter 5: Groups and homomorphisms
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5 Groups and homomorphisms
 5.1 Functions for groups

  5.1-1 Comm
  5.1-2 IsCommuting
  5.1-3 ListOfPowers
  5.1-4 GeneratorsAndInverses
  5.1-5 UpperFittingSeries
 5.2 Left Cosets for Groups

  5.2-1 LeftCoset
  5.2-2 Inverse
 5.3 Functions for group homomorphisms

  5.3-1 EpimorphismByGenerators
  5.3-2 Pullback
  5.3-3 CentralProduct
  5.3-4 IdempotentEndomorphisms
  5.3-5 DirectProductOfFunctions
  5.3-6 DirectProductOfAutomorphismGroups

5 Groups and homomorphisms

5.1 Functions for groups

5.1-1 Comm
‣ Comm( L )( operation )

This method has been transferred from package ResClasses.

It provides a method for Comm when the argument is a list (enclosed in square brackets), and calls the function LeftNormedComm.


gap> Comm( [ (1,2), (2,3) ] );
(1,2,3)
gap> Comm( [(1,2),(2,3),(3,4),(4,5),(5,6)] );
(1,5,6)
gap> Comm(Comm(Comm(Comm((1,2),(2,3)),(3,4)),(4,5)),(5,6));  ## the same
(1,5,6)

5.1-2 IsCommuting
‣ IsCommuting( a, b )( operation )

This function has been transferred from package ResClasses.

It tests whether two elements in a group commute.


gap> D12 := DihedralGroup( 12 );
<pc group of size 12 with 3 generators>
gap> SetName( D12, "D12" ); 
gap> a := D12.1;;  b := D12.2;;  
gap> IsCommuting( a, b );
false

5.1-3 ListOfPowers
‣ ListOfPowers( g, exp )( operation )

This function has been transferred from package RCWA.

The operation ListOfPowers(g,exp) returns the list [g,g^2,...,g^exp] of powers of the element g.


gap> ListOfPowers( 2, 20 );
[ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384,
 32768, 65536, 131072, 262144, 524288, 1048576 ]
gap> ListOfPowers( (1,2,3)(4,5), 12 );
[ (1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), (),
 (1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), () ]
gap> ListOfPowers( D12.2, 6 );
[ f2, f3, f2*f3, f3^2, f2*f3^2, <identity> of ... ]

5.1-4 GeneratorsAndInverses
‣ GeneratorsAndInverses( G )( operation )

This function has been transferred from package RCWA.

This operation returns a list containing the generators of G followed by the inverses of these generators.


gap> GeneratorsAndInverses( D12 );
[ f1, f2, f3, f1, f2*f3^2, f3^2 ]
gap> GeneratorsAndInverses( SymmetricGroup(5) );     
[ (1,2,3,4,5), (1,2), (1,5,4,3,2), (1,2) ]

5.1-5 UpperFittingSeries
‣ UpperFittingSeries( G )( attribute )
‣ LowerFittingSeries( G )( attribute )
‣ FittingLength( G )( attribute )

These three functions have been transferred from package ResClasses.

The upper and lower Fitting series and the Fitting length of a solvable group are described here: https://en.wikipedia.org/wiki/Fitting_length.


gap> UpperFittingSeries( D12 );  LowerFittingSeries( D12 );
[ Group([  ]), Group([ f3, f2*f3 ]), Group([ f1, f3, f2*f3 ]) ]
[ D12, Group([ f3 ]), Group([  ]) ]
gap> FittingLength( D12 );
2
gap> S4 := SymmetricGroup( 4 );;
gap> UpperFittingSeries( S4 );
[ Group(()), Group([ (1,2)(3,4), (1,4)(2,3) ]), Group([ (1,2)(3,4), (1,4)
  (2,3), (2,4,3) ]), Group([ (3,4), (2,3,4), (1,2)(3,4) ]) ]
gap> List( last, StructureDescription );
[ "1", "C2 x C2", "A4", "S4" ]
gap> LowerFittingSeries( S4 );
[ Sym( [ 1 .. 4 ] ), Alt( [ 1 .. 4 ] ), Group([ (1,4)(2,3), (1,3)
 (2,4) ]), Group(()) ]
gap> List( last, StructureDescription );
[ "S4", "A4", "C2 x C2", "1" ]
gap> FittingLength( S4);
3

5.2 Left Cosets for Groups

5.2-1 LeftCoset
‣ LeftCoset( g, U )( operation )

Since GAP uses right actions by default, the library contains the operation RightCoset(U,g) for constructing the right coset Ug of a subgroup U ≤ G and an element g ∈ G. It has been noted in the reference manual that, by inverting all the elements in Ug, the left coset g^-1U is obtained.

Just for the sake of completeness, from August 2022 this package provides the operation LeftCoset(g,U) for constructing the left coset gU. Users are strongly recommended to continue to use RightCoset for all serious calculations, since left cosets have a much simpler implementation and do not behave exactly like right cosets.

The methods for left cosets which are provided generally work by converting gU to Ug^-1; applying the equivalent method for right cosets; and, if necessary, converting back again to left cosets.

G acts on gU by OnLeftInverse: (gU)^g_0 = g_0^-1*(gU) = (g_0^-1g)U.


gap> lc1 := LeftCoset( (1,2,3), Group( [ (1,2), (3,4) ] ) ); 
LeftCoset((1,2,3),Group([ (1,2), (3,4) ]))
gap> Representative( lc1 );
(1,2,3)
gap> ActingDomain( lc1 );
Group([ (1,2), (3,4) ])
gap> AsSet( lc1 );
[ (2,3), (2,4,3), (1,2,3), (1,2,4,3) ]
gap> (1,2,3) in lc1;
true
gap> lc2 := (2,4,3) * lc1;   
LeftCoset((1,2,4),Group([ (1,2), (3,4) ]))
gap> lc3 := lc1^(2,3,4);;
gap> lc2 = lc3;            
true

5.2-2 Inverse

The inverse of the left coset gU is the right coset Ug^-1, and conversely. This is an abuse of the attribute Inverse, since the standard requirement, that x*x^-1 is an identity, does not hold.


gap> rc1 := Inverse( lc1 ); 
RightCoset(Group([ (1,2), (3,4) ]),(1,3,2))
gap> rc4 := RightCoset( Group( (1,2), (2,3) ), (3,4) ); 
RightCoset(Group([ (1,2), (2,3) ]),(3,4))
gap> lc4 := Inverse( rc4 );
LeftCoset((3,4),Group([ (1,2), (2,3) ]))
gap> Intersection( lc2, lc4 );
[ (2,3,4), (1,2,3,4) ]

5.3 Functions for group homomorphisms

5.3-1 EpimorphismByGenerators
‣ EpimorphismByGenerators( G, H )( operation )

This function has been transferred from package RCWA.

It constructs a group homomorphism which maps the generators of G to those of H. Its intended use is when G is a free group, and a warning is printed when this is not the case. Note that anything may happen if the resulting map is not a homomorphism!


gap> G := Group( (1,2,3), (3,4,5), (5,6,7), (7,8,9) );;
gap> phi := EpimorphismByGenerators( FreeGroup("a","b","c","d"), G );
[ a, b, c, d ] -> [ (1,2,3), (3,4,5), (5,6,7), (7,8,9) ]
gap> PreImagesRepresentativeNC( phi, (1,2,3,4,5,6,7,8,9) );
d*c*b*a
gap> a := G.1;; b := G.2;; c := G.3;; d := G.4;;
gap> d*c*b*a;
(1,2,3,4,5,6,7,8,9)
gap> ## note that it is easy to produce nonsense: 
gap> epi := EpimorphismByGenerators( Group((1,2,3)), Group((8,9)) );
Warning: calling GroupHomomorphismByImagesNC without checks
[ (1,2,3) ] -> [ (8,9) ]
gap> IsGroupHomomorphism( epi );
true
gap> Image( epi, (1,2,3) );                                            
()
gap> Image( epi, (1,3,2) );
(8,9)

5.3-2 Pullback
‣ Pullback( hom1, hom2 )( operation )
‣ PullbackInfo( G )( attribute )

If ϕ_1 : G_1 -> H and ϕ_2 : G_2 -> H are two group homomorphisms with the same range, then their pullback is the subgroup of G_1 × G_2 consisting of those elements (g_1,g_2) such that ϕ_1 g_1 = ϕ_2 g_2.

The attribute PullbackInfo of a pullback group P is similar to DirectProductInfo for a direct product of groups. Its value is a record with the following components:

directProduct

the direct product G_1 × G_2, and

projections

a list with the two projections onto G_1 and G_2.

There are no embeddings in this record, but it is possible to use the embeddings into the direct product, see Embedding (Reference: Embedding).


gap> s4 := Group( (1,2),(2,3),(3,4) );;
gap> s3 := Group( (5,6),(6,7) );;
gap> c3 := Subgroup( s3, [ (5,6,7) ] );;
gap> f := GroupHomomorphismByImages( s4, s3, 
>             [(1,2),(2,3),(3,4)], [(5,6),(6,7),(5,6)] );; 
gap> i := GroupHomomorphismByImages( c3, s3, [(5,6,7)], [(5,6,7)] );; 
gap> Pfi := Pullback( f, i );
Group([ (2,3,4)(5,7,6), (1,2)(3,4) ])
gap> StructureDescription( Pfi );
"A4"
gap> info := PullbackInfo( Pfi );
rec( directProduct := Group([ (1,2), (2,3), (3,4), (5,6,7) ]), 
  projections := [ [ (2,3,4)(5,7,6), (1,2)(3,4) ] -> [ (2,3,4), (1,2)(3,4) ], 
      [ (2,3,4)(5,7,6), (1,2)(3,4) ] -> [ (5,7,6), () ] ] )
gap> g := (1,2,3)(5,6,7);; 
gap> ImageElm( info!.projections[1], g );
(1,2,3)
gap> ImageElm( info!.projections[2], g );
(5,6,7) 
gap> dp := info!.directProduct;; 
gap> a := ImageElm( Embedding( dp, 1 ), (1,4,3) );; 
gap> b := ImageElm( Embedding( dp, 2 ), (5,7,6) );; 
gap> a*b in Pfi;
true

5.3-3 CentralProduct
‣ CentralProduct( G1, G2, Z1, Phi )( operation )
‣ CentralProductInfo( G )( attribute )

This function was added by Thomas Breuer, following discussions with Hongyi Zhao (see https://github.com/gap-packages/hap/issues/73).

Let G1 and G2 be two groups, Z1 be a central subgroup of G1, and Phi be an isomorphism from Z1 to a central subgroup of G2. The central product defined by these arguments is the factor group of the direct product of G1 and G2 by the central subgroup { (z, (Phi(z))^-1) : z ∈ Z1 }.

The attribute CentralProductInfo of a group G that has been created by CentralProduct is similar to PullbackInfo (5.3-2) for pullback groups. Its value is a record with the following components.

projection

the epimorphism from the direct product of G1 and G2 to G, and

phi

the map Phi.

Note that one can access the direct product as the Source (Reference: Source) value of the projection map, and one can access G1 and G2 as the two embeddings of this direct product, see Embedding (Reference: Embedding).

gap> g1 := DihedralGroup( 8 );
<pc group of size 8 with 3 generators>
gap> c1 := Centre( g1 );
Group([ f3 ])
gap> cp1 := CentralProduct( g1, g1, c1, IdentityMapping( c1 ) );
Group([ f1, f2, f5, f3, f4, f5 ])
gap> IdGroup( cp1 ) = IdGroup( ExtraspecialGroup( 2^5, "+" ) );
true
gap> g2 := QuaternionGroup( 8 );
<pc group of size 8 with 3 generators>
gap> c2 := Centre( g2 );
Group([ y2 ])
gap> cp2 := CentralProduct( g2, g2, c2, IdentityMapping( c2 ) );
Group([ f1, f2, f5, f3, f4, f5 ])
gap> IdGroup( cp2 ) = IdGroup( ExtraspecialGroup( 2^5, "+" ) );
true
gap> info2 := CentralProductInfo( cp2 );
rec( phi := IdentityMapping( Group([ y2 ]) ), 
  projection := [ f1, f2, f3, f4, f5, f6 ] -> [ f1, f2, f5, f3, f4, f5 ] )
gap> Source( Embedding( Source( info2.projection ), 1 ) ) = g2;
true

5.3-4 IdempotentEndomorphisms
‣ IdempotentEndomorphisms( G )( operation )
‣ IdempotentEndomorphismsData( G )( attribute )
‣ IdempotentEndomorphismsWithImage( genG, R )( operation )

An endomorphism f : G -> G is idempotent if f^2=f. It has an image R leqslant G; is the identity map when restricted to R; and has a kernel N which has trivial intersection with R and has size |G|/|R|.

The operation IdempotentEndomorphismsWithImage(genG,R) returns a list of the images of the generating set genG of a group G under the idempotent endomorphisms with image R.

The attribute IdempotentEndomorphismsData(G) returns a record data with fields data.gens, a fixed generating set for G, and data.images a list of the non-empty outputs of IdempotentEndomorphismsWithImage(genG,R) obtained by iterating over all subgroups of G.

The operation IdempotentEndomorphisms(G) returns the list of these mappings obtained using IdempotentEndomorphismsData(G). The first of these is the zero map, the second is the identity.


gap> gens := [ (1,2,3,4), (1,2)(3,4) ];; 
gap> d8 := Group( gens );;
gap> SetName( d8, "d8" );
gap> c2 := Subgroup( d8, [ (2,4) ] );;
gap> IdempotentEndomorphismsWithImage( gens, c2 );
[ [ (), (2,4) ], [ (2,4), () ] ]
gap> IdempotentEndomorphismsData( d8 );
rec( gens := [ (1,2,3,4), (1,2)(3,4) ], 
  images := [ [ [ (), () ] ], [ [ (), (2,4) ], [ (2,4), () ] ], 
      [ [ (), (1,3) ], [ (1,3), () ] ], 
      [ [ (), (1,2)(3,4) ], [ (1,2)(3,4), (1,2)(3,4) ] ], 
      [ [ (), (1,4)(2,3) ], [ (1,4)(2,3), (1,4)(2,3) ] ], 
      [ [ (1,2,3,4), (1,2)(3,4) ] ] ] )
gap> List( last.images, L -> Length(L) );
[ 1, 2, 2, 2, 2, 1 ]
gap> IdempotentEndomorphisms( d8 );               
[ [ (1,2,3,4), (1,2)(3,4) ] -> [ (), () ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (2,4) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (2,4), () ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (1,3) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,3), () ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (1,2)(3,4) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,2)(3,4), (1,2)(3,4) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (1,4)(2,3) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,4)(2,3), (1,4)(2,3) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,2,3,4), (1,2)(3,4) ] ]

The quaternion group q8 is an example of a group with a tail: there is only one subgroup in the lattice which covers the identity subgroup. The only idempotent isomorphisms of such groups are the identity mapping and the zero mapping because the only pairs N,R are the whole group and the identity subgroup.


gap> q8 := QuaternionGroup( 8 );;
gap> IdempotentEndomorphisms( q8 );
[ [ x, y ] -> [ <identity> of ..., <identity> of ... ], [ x, y ] -> [ x, y ] ]

5.3-5 DirectProductOfFunctions
‣ DirectProductOfFunctions( G, H, f1, f2 )( operation )

Given group homomorphisms f_1 : G_1 -> G_2 and f_2 : H_1 -> H_2, this operation return the product homomorphism f_1 × f_2 : G_1 × G_2 -> H_1 × H_2.


gap> c4 := Group( (1,2,3,4) );; 
gap> c2 := Group( (5,6) );; 
gap> f1 := GroupHomomorphismByImages( c4, c2, [(1,2,3,4)], [(5,6)] );;
gap> c3 := Group( (1,2,3) );; 
gap> c6 := Group( (1,2,3,4,5,6) );; 
gap> f2 := GroupHomomorphismByImages( c3, c6, [(1,2,3)], [(1,3,5)(2,4,6)] );; 
gap> c4c3 := DirectProduct( c4, c3 ); 
Group([ (1,2,3,4), (5,6,7) ])
gap> c2c6 := DirectProduct( c2, c6 ); 
Group([ (1,2), (3,4,5,6,7,8) ])
gap> f := DirectProductOfFunctions( c4c3, c2c6, f1, f2 ); 
[ (1,2,3,4), (5,6,7) ] -> [ (1,2), (3,5,7)(4,6,8) ]
gap> ImageElm( f, (1,4,3,2)(5,7,6) ); 
(1,2)(3,7,5)(4,8,6)

5.3-6 DirectProductOfAutomorphismGroups
‣ DirectProductOfAutomorphismGroups( A1, A2 )( operation )

Let A_1,A_2 be groups of automorphism of groups G_1,G_2 respectively. The output of this function is a group A_1 × A_2 of automorphisms of G_1 × G_2.


gap> c9 := Group( (1,2,3,4,5,6,7,8,9) );; 
gap> ac9 := AutomorphismGroup( c9 );; 
gap> q8 := QuaternionGroup( IsPermGroup, 8 );;
gap> aq8 := AutomorphismGroup( q8 );;
gap> A := DirectProductOfAutomorphismGroups( ac9, aq8 );
<group with 5 generators>
gap> genA := GeneratorsOfGroup( A );;
gap> G := Source( genA[1] );
Group([ (1,2,3,4,5,6,7,8,9), (10,14,12,16)(11,17,13,15), (10,11,12,13)
(14,15,16,17) ])
gap> a := genA[1]*genA[5];  
[ (1,2,3,4,5,6,7,8,9), (10,14,12,16)(11,17,13,15), (10,11,12,13)(14,15,16,17) 
 ] -> [ (1,3,5,7,9,2,4,6,8), (10,16,12,14)(11,15,13,17), 
  (10,11,12,13)(14,15,16,17) ]
gap> ImageElm( a, (1,9,8,7,6,5,4,3,2)(10,14,12,16)(11,17,13,15) );
(1,8,6,4,2,9,7,5,3)(10,16,12,14)(11,15,13,17)
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utils-0.86/doc/transfer.xml000644 000765 000024 00000011463 14621444241 021035 0ustar00christopherwensleystaff000000 000000 The transfer procedure We consider here the process for transferring utility functions from a package &Home; to &Utils; which has to avoid the potential problem of duplicate declarations of a function causing loading problems in &GAP;.

If the functions in &Home; all have names of the form HOME_FunctionName then, in &Utils;, these functions are likely to be renamed as FunctionName or something similar. In this case the problem of duplicate declarations does not arise. This is what has happened with transfers from the &AutoDoc; package.

The case where the function names are unchanged is more complicated. Initially we tried out a process which allowed repeated declarations and installations of the functions being transferred. This involved additions to the main library files global.g and oper.g. Since there were misgivings about interfering in this way with basic operations such as BIND_GLOBAL, a simpler (but slightly less convenient) process has been adopted.

Using this alternative procedure, the following steps will be followed when making transfers from &Home; to &Utils;. (&Home;:) Offer functions for inclusion. This may be simply done by emailing a list of functions. More usefully, email the declaration, implementation, test and documentation files, e.g.: home.gd, home.gi, home.tst and home.xml. (All active authors should be involved.) (&Home;:) Declare that m.n is the last version of &Home; to contain these functions, so that m.n+1 (or similar) will be the first version of &Home; to have all these functions removed, and to specify &Utils; as a required package. (&Utils;:) Add strings "home" and "m.n" to the list UtilsPackageVersions in the file utils/lib/start.gd. While the transfers are being made, it is essential that any new versions of &Home; should be tested with the latest version of &Utils; before they are released, so as to avoid loading failures. (&Utils;:) Include the function declaration and implementation sections in suitable files, enclosed within a conditional clause of the form: . . . . . . fi; ]]> OKtoReadFromUtils The function OKtoReadFromUtils returns true only if there is an installed version of &Home; and if this version is greater than m.n. So, at this stage, the copied code will not be read, and the transferred functions can only be called if &Home; has been installed. (&Utils;:) Add the test and documentation material to the appropriate files. The copied code can be tested by temporarily moving &Home; away from &GAP;'s package directory. (&Utils;:) Release a new version of &Utils; containing all the transferred material. (&Home;:) Edit out the declarations and implementations of all the transferred functions, and remove references to them in the manual and tests. Possibly add a note to the manual that these functions have been transferred. Add &Utils; to the list of &Home;'s required packages in PackageInfo.g. Release a new version of &Home;. (&Utils;:) In due course, when the new version(s) of &Home; are well established, it may be safe to remove the conditional clauses mentioned in item 4 above. The entry for &Home; in UtilsPackageLists may then be removed.

Finally, a note on the procedure for testing these functions. As long as a function being transferred still exists in the &Home; package, the code will not be read from &Utils;. So, when the tests are run, it is necessary to LoadPackage("home") before the function is called. 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6 Matrices
 6.1 Some operations for matrices

  6.1-1 DirectSumDecompositionMatrices

6 Matrices

6.1 Some operations for matrices

6.1-1 DirectSumDecompositionMatrices
‣ DirectSumDecompositionMatrices( M )( operation )

In June 2023 Hongyi Zhao asked in the Forum for a function to implement matrix decomposition into blocks. Such a function was then provided by Pedro García-Sánchez. Hongyi Zhao then requested that the function be added to Utils. What is provided here is a revised version of the original solution, returning a list of decompositions.

This function is a partial inverse to the undocumented library operation DirectSumMat. So if L is the list of diagonal decompositions of a matrix M then each entry in L is a list of matrices, and the direct sum of each of these lists is equal to the original M.

In the following examples, M_6 is an obvious direct sum with 3 blocks. M_4 is an example with three decompositions, while M_8 = M_4 ⊕ M_4 has 16 decompositions (not listed).


gap> M6 := [ [1,2,0,0,0,0], [3,4,0,0,0,0], [5,6,0,0,0,0],                       
>            [0,0,9,0,0,0], [0,0,0,1,2,3], [0,0,0,4,5,6] ];;
gap> Display( M6 );
[ [  1,  2,  0,  0,  0,  0 ],
  [  3,  4,  0,  0,  0,  0 ],
  [  5,  6,  0,  0,  0,  0 ],
  [  0,  0,  9,  0,  0,  0 ],
  [  0,  0,  0,  1,  2,  3 ],
  [  0,  0,  0,  4,  5,  6 ] ]
gap> L6 := DirectSumDecompositionMatrices( M6 );
[ [ [ [ 1, 2 ], [ 3, 4 ], [ 5, 6 ] ], [ [ 9 ] ], [ [ 1, 2, 3 ], [ 4, 5, 6 ] ] 
     ] ]

gap> M4 := [ [0,3,0,0], [0,0,0,0], [0,0,0,0], [0,0,4,0] ];;
gap> Display( M4 );
[ [  0,  3,  0,  0 ],
  [  0,  0,  0,  0 ],
  [  0,  0,  0,  0 ],
  [  0,  0,  4,  0 ] ]
gap> L4 := DirectSumDecompositionMatrices( M4 );
[ [ [ [ 0, 3 ] ], [ [ 0, 0 ], [ 0, 0 ], [ 4, 0 ] ] ], 
  [ [ [ 0, 3 ], [ 0, 0 ] ], [ [ 0, 0 ], [ 4, 0 ] ] ], 
  [ [ [ 0, 3 ], [ 0, 0 ], [ 0, 0 ] ], [ [ 4, 0 ] ] ] ]
gap> for L in L4 do 
>        A := DirectSumMat( L );; 
>        if ( A = M4 ) then Print( "yes, A = M4\n" ); fi; 
>    od;
yes, A = M4
yes, A = M4
yes, A = M4

gap> M8 := DirectSumMat( M4, M4 );; 
gap> Display( M8 );
[ [  0,  3,  0,  0,  0,  0,  0,  0 ],
  [  0,  0,  0,  0,  0,  0,  0,  0 ],
  [  0,  0,  0,  0,  0,  0,  0,  0 ],
  [  0,  0,  4,  0,  0,  0,  0,  0 ],
  [  0,  0,  0,  0,  0,  3,  0,  0 ],
  [  0,  0,  0,  0,  0,  0,  0,  0 ],
  [  0,  0,  0,  0,  0,  0,  0,  0 ],
  [  0,  0,  0,  0,  0,  0,  4,  0 ] ]
gap> L8 := DirectSumDecompositionMatrices( M8 );;
gap> Length( L8 ); 
16

The current method does not, however, catch all possible decompositions. In the following example the matrix M_5 has its third row and third column extirely zero, and the only decomposition found has a [0] factor. There are clearly two 2-factor decompositions with a 2-by-3 and a 3-by-2 factor, but these are not found at present.


gap> M5 := [ [1,2,0,0,0], [3,4,0,0,0], [0,0,0,0,0],
>            [0,0,0,6,7], [0,0,0,8,9] ];;
gap> Display(M5);
[ [  1,  2,  0,  0,  0 ],
  [  3,  4,  0,  0,  0 ],
  [  0,  0,  0,  0,  0 ],
  [  0,  0,  0,  6,  7 ],
  [  0,  0,  0,  8,  9 ] ]
gap> L5 := DirectSumDecompositionMatrices( M5 ); 
[ [ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 0 ] ], [ [ 6, 7 ], [ 8, 9 ] ] ] ]
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utils-0.86/doc/print.xml000644 000765 000024 00000004427 14621444241 020347 0ustar00christopherwensleystaff000000 000000 Printing Lists and Iterators
Printing selected items The functions described here print lists or objects with an iterator with one item per line, either the whole list/iterator or certain subsets: by giving a list of positions of items to be printed, or by specifying a first item and then a regular step. This function, given three (or four) parameters, calls operations PrintSelectionFromList or PrintSelectionFromIterator which prints the first item specified, and then the item at every step. The fourth parameter is essential when the object being printed is infinite.

Alternatively, given two parameters, with the second parameter a list L of positive integers, only the items at positions in L are printed.

L := List( [1..20], n -> n^5 );; gap> PrintSelection( L, [18..20] ); 18 : 1889568 19 : 2476099 20 : 3200000 gap> PrintSelection( L, 2, 9 ); 2 : 32 11 : 161051 20 : 3200000 gap> PrintSelection( L, 2, 3, 11 ); 2 : 32 5 : 3125 8 : 32768 11 : 161051 gap> s5 := SymmetricGroup( 5 );; gap> PrintSelection( s5, [30,31,100,101] ); 30 : (1,5)(3,4) 31 : (1,5,2) 100 : (1,4,3) 101 : (1,4)(3,5) gap> PrintSelection( s5, 1, 30 ); 1 : () 31 : (1,5,2) 61 : (1,2,3) 91 : (1,3,5,2,4) gap> PrintSelection( s5, 9, 11, 43 ); 9 : (2,5,3) 20 : (2,4) 31 : (1,5,2) 42 : (1,5,2,3,4) ]]>

utils-0.86/doc/chapBib.txt000644 000765 000024 00000003411 14621610512 020546 0ustar00christopherwensleystaff000000 000000 References [BCP97] Bosma, W., Cannon, J. and Playoust, C., The Magma algebra system. {I}. The user language, J. Symbolic Comput., 24, 3-4 (1997), 235--265 pages, ( Computational algebra and number theory (London, 1993)} https://doi.org/10.1006/jsco.1996.0125 ). [GH16] Gutsche, S. and Horn, M., AutoDoc - Generate documentation from GAP source code (Version 2016.12.04) (2016), ( GAP package, https://github.com/gap-packages/AutoDoc ). [Hor17] Horn, M., GitHubPagesForGAP - Template for easily using GitHub Pages within GAP packages (Version 0.2) (2017), ( GAP package, https://gap-system.github.io/GitHubPagesForGAP/ ). [Koh17a] Kohl, S., RCWA - Residue-Class-Wise Affine Groups (Version 4.5.1) (2017), ( GAP package, https://stefan-kohl.github.io/rcwa.html ). [Koh17b] Kohl, S., ResClasses - Set-Theoretic Computations with Residue Classes (Version 4.6.0) (2017), ( GAP package, https://stefan-kohl.github.io/resclasses.html ). [LN17] Lübeck, F. and Neunhöffer, M., GAPDoc (Version 1.6), RWTH Aachen (2017), ( GAP package, https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/index.html ). [WAOU17] Wensley, C. D., Alp, M., Odabas, A. and Uslu, E. O., XMod - Crossed Modules and Cat1-groups in GAP (Version 2.64) (2017), ( GAP package, https://github.com/gap-packages/xmod ).  utils-0.86/doc/chap7_mj.html000644 000765 000024 00000027310 14621610515 021042 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Chapter 7: Iterators
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7 Iterators
 7.1 Some iterators for groups and their isomorphisms

  7.1-1 AllIsomorphismsIterator
  7.1-2 AllSubgroupsIterator
 7.2 Operations on iterators

  7.2-1 CartesianIterator
  7.2-2 UnorderedPairsIterator

7 Iterators

7.1 Some iterators for groups and their isomorphisms

The motivation for adding these operations is partly to give a simple example of an iterator for a list that does not yet exist, and need not be created.

7.1-1 AllIsomorphismsIterator
‣ AllIsomorphismsIterator( G, H )( operation )
‣ AllIsomorphismsNumber( G, H )( operation )
‣ AllIsomorphisms( G, H )( operation )

The main GAP library contains functions producing complete lists of group homomorphisms such as AllHomomorphisms; AllEndomorphisms and AllAutomorphisms. Here we add the missing AllIsomorphisms(G,H) for a list of isomorphisms from \(G\) to \(H\). The method is simple -- find one isomorphism \(G \to H\) and compose this with all the automorphisms of \(G\). In all these cases it may not be desirable to construct a list of homomorphisms, but just implement an iterator, and that is what is done here. The operation AllIsomorphismsNumber returns the number of isomorphisms iterated over (this is, of course, just the order of the automorphisms group). The operation AllIsomorphisms produces the list or isomorphisms.


gap> G := SmallGroup( 6,1);; 
gap> iter := AllIsomorphismsIterator( G, s3 );;
gap> NextIterator( iter );
[ f1, f2 ] -> [ (6,7), (5,6,7) ]
gap> n := AllIsomorphismsNumber( G, s3 );
6
gap> AllIsomorphisms( G, s3 );
[ [ f1, f2 ] -> [ (6,7), (5,6,7) ], [ f1, f2 ] -> [ (5,7), (5,6,7) ], 
  [ f1, f2 ] -> [ (5,6), (5,7,6) ], [ f1, f2 ] -> [ (6,7), (5,7,6) ], 
  [ f1, f2 ] -> [ (5,7), (5,7,6) ], [ f1, f2 ] -> [ (5,6), (5,6,7) ] ]
gap> iter := AllIsomorphismsIterator( G, s3 );;
gap> for h in iter do Print( ImageElm( h, G.1 ) = (6,7), ", " ); od;
true, false, false, true, false, false,

7.1-2 AllSubgroupsIterator
‣ AllSubgroupsIterator( G )( operation )

The manual entry for the operation AllSubgroups states that it is only intended to be used on small examples in a classroom situation. Access to all subgroups was required by the XMod package, so this iterator was introduced here. It used the operations LatticeSubgroups(G) and ConjugacyClassesSubgroups(lat), and then iterates over the entries in these classes.


gap> c3c3 := Group( (1,2,3), (4,5,6) );; 
gap> iter := AllSubgroupsIterator( c3c3 );
<iterator>
gap> while not IsDoneIterator(iter) do Print(NextIterator(iter),"\n"); od;
Group( () )
Group( [ (4,5,6) ] )
Group( [ (1,2,3) ] )
Group( [ (1,2,3)(4,5,6) ] )
Group( [ (1,3,2)(4,5,6) ] )
Group( [ (4,5,6), (1,2,3) ] )

7.2 Operations on iterators

This section considers ways of producing an iterator from one or more iterators. It may be that operations equivalent to these are available elsewhere in the library -- if so, the ones here can be removed in due course.

7.2-1 CartesianIterator
‣ CartesianIterator( iter1, iter2 )( operation )

This iterator returns all pairs \([x,y]\) where \(x\) is the output of a first iterator and \(y\) is the output of a second iterator.


gap> it1 := Iterator( [ 1, 2, 3 ] );;
gap> it2 := Iterator( [ 4, 5, 6 ] );;
gap> iter := CartesianIterator( it1, it2 );;
gap> while not IsDoneIterator(iter) do Print(NextIterator(iter),"\n"); od;
[ 1, 4 ]
[ 1, 5 ]
[ 1, 6 ]
[ 2, 4 ]
[ 2, 5 ]
[ 2, 6 ]
[ 3, 4 ]
[ 3, 5 ]
[ 3, 6 ]

7.2-2 UnorderedPairsIterator
‣ UnorderedPairsIterator( iter )( operation )

This operation returns pairs \([x,y]\) where \(x,y\) are output from a given iterator iter. Unlike the output from CartesianIterator(iter,iter), unordered pairs are returned. In the case \(L = [1,2,3,\ldots]\) the pairs are ordered as \([1,1],[1,2],[2,2],[1,3],[2,3],[3,3],\ldots\).


gap> L := [6,7,8,9];;
gap> iterL := IteratorList( L );; 
gap> pairsL := UnorderedPairsIterator( iterL );;                              
gap> while not IsDoneIterator(pairsL) do Print(NextIterator(pairsL),"\n"); od;
[ 6, 6 ]
[ 6, 7 ]
[ 7, 7 ]
[ 6, 8 ]
[ 7, 8 ]
[ 8, 8 ]
[ 6, 9 ]
[ 7, 9 ]
[ 8, 9 ]
[ 9, 9 ]
gap> iter4 := IteratorList( [ 4 ] );
<iterator>
gap> pairs4 := UnorderedPairsIterator(iter4);
<iterator>
gap> NextIterator( pairs4 );
[ 4, 4 ]
gap> IsDoneIterator( pairs4 );
true
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utils-0.86/doc/lists.xml000644 000765 000024 00000016465 14621444241 020356 0ustar00christopherwensleystaff000000 000000 Lists, Sets and Strings
Functions for lists This function has been transferred from package &ResClasses;.

It takes a list L of length n and outputs the list of length n-1 containing all the differences L[i]-L[i-1].

List( [1..12], n->n^3 ); [ 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 ] gap> DifferencesList( last ); [ 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397 ] gap> DifferencesList( last ); [ 12, 18, 24, 30, 36, 42, 48, 54, 60, 66 ] gap> DifferencesList( last ); [ 6, 6, 6, 6, 6, 6, 6, 6, 6 ] ]]> These functions have been transferred from package &ResClasses;.

They take a list L of length n and output the quotients L[i]/L[i-1] of consecutive entries in L. An error is returned if an entry is zero.

List( [0..10], n -> Factorial(n) ); [ 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 ] gap> QuotientsList( last ); [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ] gap> L := [ 1, 3, 5, -1, -3, -5 ];; gap> QuotientsList( L ); [ 3, 5/3, -1/5, 3, 5/3 ] gap> FloatQuotientsList( L ); [ 3., 1.66667, -0.2, 3., 1.66667 ] gap> QuotientsList( [ 2, 1, 0, -1, -2 ] ); [ 1/2, 0, fail, 2 ] gap> FloatQuotientsList( [1..10] ); [ 2., 1.5, 1.33333, 1.25, 1.2, 1.16667, 1.14286, 1.125, 1.11111 ] gap> Product( last ); 10. ]]> This function has been transferred from package &RCWA;.

SearchCycle is a tool to find likely cycles in lists. What, precisely, a cycle is, is deliberately fuzzy here, and may possibly even change. The idea is that the beginning of the list may be anything, following that the same pattern needs to be repeated several times in order to be recognized as a cycle.

L := [1..20];; L[1]:=13;; gap> for i in [1..19] do > if IsOddInt(L[i]) then L[i+1]:=3*L[i]+1; else L[i+1]:=L[i]/2; fi; > od; gap> L; [ 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4 ] gap> SearchCycle( L ); [ 1, 4, 2 ] gap> n := 1;; L := [n];; gap> for i in [1..100] do n:=(n^2+1) mod 1093; Add(L,n); od; gap> L; [ 1, 2, 5, 26, 677, 363, 610, 481, 739, 715, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004 ] gap> C := SearchCycle( L ); [ 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754 ] gap> P := Positions( L, 157 ); [ 14, 26, 38, 50, 62, 74, 86, 98 ] gap> Length( C ); DifferencesList( P ); 12 [ 12, 12, 12, 12, 12, 12, 12 ] ]]> This function has been transferred from package &ResClasses;.

It returns a random unordered k-tuple of distinct elements of a set S.

## "6 aus 49" is a common lottery in Germany gap> RandomCombination( [1..49], 6 ); [ 2, 16, 24, 26, 37, 47 ] ]]>

Distinct and Common Representatives distinct and common representatives These operations have been transferred from package &XMod;.

They deal with lists of subsets of [1 \ldots n] and construct systems of distinct and common representatives using simple, non-recursive, combinatorial algorithms.

When L is a set of n subsets of [1 \ldots n] and the Hall condition is satisfied (the union of any k subsets has at least k elements), a set of DistinctRepresentatives exists.

When J,K are both lists of n sets, the operation CommonRepresentatives returns two lists: the set of representatives, and a permutation of the subsets of the second list.

The operation CommonTransversal may be used to provide a common transversal for the sets of left and right cosets of a subgroup H of a group G, although a greedy algorithm is usually quicker.

J := [ [1,2,3], [3,4], [3,4], [1,2,4] ];; gap> DistinctRepresentatives( J ); [ 1, 3, 4, 2 ] gap> K := [ [3,4], [1,2], [2,3], [2,3,4] ];; gap> CommonRepresentatives( J, K ); [ [ 3, 3, 3, 1 ], [ 1, 3, 4, 2 ] ] gap> d16 := DihedralGroup( IsPermGroup, 16 ); Group([ (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) ]) gap> SetName( d16, "d16" ); gap> c4 := Subgroup( d16, [ d16.1^2 ] ); Group([ (1,3,5,7)(2,4,6,8) ]) gap> SetName( c4, "c4" ); gap> RightCosets( d16, c4 ); [ RightCoset(c4,()), RightCoset(c4,(2,8)(3,7)(4,6)), RightCoset(c4,(1,8,7,6,5, 4,3,2)), RightCoset(c4,(1,8)(2,7)(3,6)(4,5)) ] gap> trans := CommonTransversal( d16, c4 ); [ (), (2,8)(3,7)(4,6), (1,2,3,4,5,6,7,8), (1,2)(3,8)(4,7)(5,6) ] gap> IsCommonTransversal( d16, c4, trans ); true ]]>

Functions for strings This function has been transferred from package &ResClasses;.

The result of BlankFreeString( obj ); is a composite of the functions String( obj ) and RemoveCharacters( obj, " " );.

gens := GeneratorsOfGroup( DihedralGroup(12) ); [ f1, f2, f3 ] gap> String( gens ); "[ f1, f2, f3 ]" gap> BlankFreeString( gens ); "[f1,f2,f3]" ]]>

utils-0.86/doc/rainbow.js000644 000765 000024 00000005336 14621610515 020467 0ustar00christopherwensleystaff000000 000000 function randchar(str) { var i = Math.floor(Math.random() * str.length); while (i == str.length) i = Math.floor(Math.random() * str.length); return str[i]; } hexdigits = "0123456789abcdef"; function randlight() { return randchar("cdef")+randchar(hexdigits)+ randchar("cdef")+randchar(hexdigits)+ randchar("cdef")+randchar(hexdigits) } function randdark() { return randchar("012345789")+randchar(hexdigits)+ randchar("012345789")+randchar(hexdigits)+ randchar("102345789")+randchar(hexdigits) } document.write('\n'); utils-0.86/doc/chap12_mj.html000644 000765 000024 00000020274 14621610515 021120 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Chapter 12: The transfer procedure
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12 The transfer procedure

12 The transfer procedure

We consider here the process for transferring utility functions from a package Home to Utils which has to avoid the potential problem of duplicate declarations of a function causing loading problems in GAP.

If the functions in Home all have names of the form HOME_FunctionName then, in Utils, these functions are likely to be renamed as FunctionName or something similar. In this case the problem of duplicate declarations does not arise. This is what has happened with transfers from the AutoDoc package.

The case where the function names are unchanged is more complicated. Initially we tried out a process which allowed repeated declarations and installations of the functions being transferred. This involved additions to the main library files global.g and oper.g. Since there were misgivings about interfering in this way with basic operations such as BIND_GLOBAL, a simpler (but slightly less convenient) process has been adopted.

Using this alternative procedure, the following steps will be followed when making transfers from Home to Utils.

  1. (Home:) Offer functions for inclusion. This may be simply done by emailing a list of functions. More usefully, email the declaration, implementation, test and documentation files, e.g.: home.gd, home.gi, home.tst and home.xml. (All active authors should be involved.)

  2. (Home:) Declare that m.n is the last version of Home to contain these functions, so that m.n+1 (or similar) will be the first version of Home to have all these functions removed, and to specify Utils as a required package.

  3. (Utils:) Add strings "home" and "m.n" to the list UtilsPackageVersions in the file utils/lib/start.gd.

    
UtilsPackageVersions := 
  [ "autodoc",     "2016.01.31", 
    "resclasses",  "4.2.5", 
    "home",        "m.n",
    ...,           ... 
  ];

    While the transfers are being made, it is essential that any new versions of Home should be tested with the latest version of Utils before they are released, so as to avoid loading failures.

  4. (Utils:) Include the function declaration and implementation sections in suitable files, enclosed within a conditional clause of the form:

    
if OKtoReadFromUtils( "Home" ) then
. . . . . . 
 <the code> 
. . . . . . 
fi;

    The function OKtoReadFromUtils returns true only if there is an installed version of Home and if this version is greater than m.n. So, at this stage, the copied code will not be read, and the transferred functions can only be called if Home has been installed.

  5. (Utils:) Add the test and documentation material to the appropriate files. The copied code can be tested by temporarily moving Home away from GAP's package directory.

  6. (Utils:) Release a new version of Utils containing all the transferred material.

  7. (Home:) Edit out the declarations and implementations of all the transferred functions, and remove references to them in the manual and tests. Possibly add a note to the manual that these functions have been transferred. Add Utils to the list of Home's required packages in PackageInfo.g. Release a new version of Home.

  8. (Utils:) In due course, when the new version(s) of Home are well established, it may be safe to remove the conditional clauses mentioned in item 4 above. The entry for Home in UtilsPackageLists may then be removed.

Finally, a note on the procedure for testing these functions. As long as a function being transferred still exists in the Home package, the code will not be read from Utils. So, when the tests are run, it is necessary to LoadPackage("home") before the function is called. The file utils/tst/testall.g makes sure that all the necessary packages are loaded before the individual tests are called.

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utils-0.86/doc/chap6_mj.html000644 000765 000024 00000020634 14621610515 021043 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Chapter 6: Matrices
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6 Matrices
 6.1 Some operations for matrices

  6.1-1 DirectSumDecompositionMatrices

6 Matrices

6.1 Some operations for matrices

6.1-1 DirectSumDecompositionMatrices
‣ DirectSumDecompositionMatrices( M )( operation )

In June 2023 Hongyi Zhao asked in the Forum for a function to implement matrix decomposition into blocks. Such a function was then provided by Pedro García-Sánchez. Hongyi Zhao then requested that the function be added to Utils. What is provided here is a revised version of the original solution, returning a list of decompositions.

This function is a partial inverse to the undocumented library operation DirectSumMat. So if \(L\) is the list of diagonal decompositions of a matrix \(M\) then each entry in \(L\) is a list of matrices, and the direct sum of each of these lists is equal to the original \(M\).

In the following examples, \(M_6\) is an obvious direct sum with \(3\) blocks. \(M_4\) is an example with three decompositions, while \(M_8 = M_4 \oplus M_4\) has \(16\) decompositions (not listed).


gap> M6 := [ [1,2,0,0,0,0], [3,4,0,0,0,0], [5,6,0,0,0,0],                       
>            [0,0,9,0,0,0], [0,0,0,1,2,3], [0,0,0,4,5,6] ];;
gap> Display( M6 );
[ [  1,  2,  0,  0,  0,  0 ],
  [  3,  4,  0,  0,  0,  0 ],
  [  5,  6,  0,  0,  0,  0 ],
  [  0,  0,  9,  0,  0,  0 ],
  [  0,  0,  0,  1,  2,  3 ],
  [  0,  0,  0,  4,  5,  6 ] ]
gap> L6 := DirectSumDecompositionMatrices( M6 );
[ [ [ [ 1, 2 ], [ 3, 4 ], [ 5, 6 ] ], [ [ 9 ] ], [ [ 1, 2, 3 ], [ 4, 5, 6 ] ] 
     ] ]

gap> M4 := [ [0,3,0,0], [0,0,0,0], [0,0,0,0], [0,0,4,0] ];;
gap> Display( M4 );
[ [  0,  3,  0,  0 ],
  [  0,  0,  0,  0 ],
  [  0,  0,  0,  0 ],
  [  0,  0,  4,  0 ] ]
gap> L4 := DirectSumDecompositionMatrices( M4 );
[ [ [ [ 0, 3 ] ], [ [ 0, 0 ], [ 0, 0 ], [ 4, 0 ] ] ], 
  [ [ [ 0, 3 ], [ 0, 0 ] ], [ [ 0, 0 ], [ 4, 0 ] ] ], 
  [ [ [ 0, 3 ], [ 0, 0 ], [ 0, 0 ] ], [ [ 4, 0 ] ] ] ]
gap> for L in L4 do 
>        A := DirectSumMat( L );; 
>        if ( A = M4 ) then Print( "yes, A = M4\n" ); fi; 
>    od;
yes, A = M4
yes, A = M4
yes, A = M4

gap> M8 := DirectSumMat( M4, M4 );; 
gap> Display( M8 );
[ [  0,  3,  0,  0,  0,  0,  0,  0 ],
  [  0,  0,  0,  0,  0,  0,  0,  0 ],
  [  0,  0,  0,  0,  0,  0,  0,  0 ],
  [  0,  0,  4,  0,  0,  0,  0,  0 ],
  [  0,  0,  0,  0,  0,  3,  0,  0 ],
  [  0,  0,  0,  0,  0,  0,  0,  0 ],
  [  0,  0,  0,  0,  0,  0,  0,  0 ],
  [  0,  0,  0,  0,  0,  0,  4,  0 ] ]
gap> L8 := DirectSumDecompositionMatrices( M8 );;
gap> Length( L8 ); 
16

The current method does not, however, catch all possible decompositions. In the following example the matrix \(M_5\) has its third row and third column extirely zero, and the only decomposition found has a \([0]\) factor. There are clearly two \(2\)-factor decompositions with a \(2\)-by-\(3\) and a \(3\)-by-\(2\) factor, but these are not found at present.


gap> M5 := [ [1,2,0,0,0], [3,4,0,0,0], [0,0,0,0,0],
>            [0,0,0,6,7], [0,0,0,8,9] ];;
gap> Display(M5);
[ [  1,  2,  0,  0,  0 ],
  [  3,  4,  0,  0,  0 ],
  [  0,  0,  0,  0,  0 ],
  [  0,  0,  0,  6,  7 ],
  [  0,  0,  0,  8,  9 ] ]
gap> L5 := DirectSumDecompositionMatrices( M5 ); 
[ [ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 0 ] ], [ [ 6, 7 ], [ 8, 9 ] ] ] ]
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7 Iterators
 7.1 Some iterators for groups and their isomorphisms

  7.1-1 AllIsomorphismsIterator
  7.1-2 AllSubgroupsIterator
 7.2 Operations on iterators

  7.2-1 CartesianIterator
  7.2-2 UnorderedPairsIterator

7 Iterators

7.1 Some iterators for groups and their isomorphisms

The motivation for adding these operations is partly to give a simple example of an iterator for a list that does not yet exist, and need not be created.

7.1-1 AllIsomorphismsIterator
‣ AllIsomorphismsIterator( G, H )( operation )
‣ AllIsomorphismsNumber( G, H )( operation )
‣ AllIsomorphisms( G, H )( operation )

The main GAP library contains functions producing complete lists of group homomorphisms such as AllHomomorphisms; AllEndomorphisms and AllAutomorphisms. Here we add the missing AllIsomorphisms(G,H) for a list of isomorphisms from G to H. The method is simple -- find one isomorphism G -> H and compose this with all the automorphisms of G. In all these cases it may not be desirable to construct a list of homomorphisms, but just implement an iterator, and that is what is done here. The operation AllIsomorphismsNumber returns the number of isomorphisms iterated over (this is, of course, just the order of the automorphisms group). The operation AllIsomorphisms produces the list or isomorphisms.


gap> G := SmallGroup( 6,1);; 
gap> iter := AllIsomorphismsIterator( G, s3 );;
gap> NextIterator( iter );
[ f1, f2 ] -> [ (6,7), (5,6,7) ]
gap> n := AllIsomorphismsNumber( G, s3 );
6
gap> AllIsomorphisms( G, s3 );
[ [ f1, f2 ] -> [ (6,7), (5,6,7) ], [ f1, f2 ] -> [ (5,7), (5,6,7) ], 
  [ f1, f2 ] -> [ (5,6), (5,7,6) ], [ f1, f2 ] -> [ (6,7), (5,7,6) ], 
  [ f1, f2 ] -> [ (5,7), (5,7,6) ], [ f1, f2 ] -> [ (5,6), (5,6,7) ] ]
gap> iter := AllIsomorphismsIterator( G, s3 );;
gap> for h in iter do Print( ImageElm( h, G.1 ) = (6,7), ", " ); od;
true, false, false, true, false, false,

7.1-2 AllSubgroupsIterator
‣ AllSubgroupsIterator( G )( operation )

The manual entry for the operation AllSubgroups states that it is only intended to be used on small examples in a classroom situation. Access to all subgroups was required by the XMod package, so this iterator was introduced here. It used the operations LatticeSubgroups(G) and ConjugacyClassesSubgroups(lat), and then iterates over the entries in these classes.


gap> c3c3 := Group( (1,2,3), (4,5,6) );; 
gap> iter := AllSubgroupsIterator( c3c3 );
<iterator>
gap> while not IsDoneIterator(iter) do Print(NextIterator(iter),"\n"); od;
Group( () )
Group( [ (4,5,6) ] )
Group( [ (1,2,3) ] )
Group( [ (1,2,3)(4,5,6) ] )
Group( [ (1,3,2)(4,5,6) ] )
Group( [ (4,5,6), (1,2,3) ] )

7.2 Operations on iterators

This section considers ways of producing an iterator from one or more iterators. It may be that operations equivalent to these are available elsewhere in the library -- if so, the ones here can be removed in due course.

7.2-1 CartesianIterator
‣ CartesianIterator( iter1, iter2 )( operation )

This iterator returns all pairs [x,y] where x is the output of a first iterator and y is the output of a second iterator.


gap> it1 := Iterator( [ 1, 2, 3 ] );;
gap> it2 := Iterator( [ 4, 5, 6 ] );;
gap> iter := CartesianIterator( it1, it2 );;
gap> while not IsDoneIterator(iter) do Print(NextIterator(iter),"\n"); od;
[ 1, 4 ]
[ 1, 5 ]
[ 1, 6 ]
[ 2, 4 ]
[ 2, 5 ]
[ 2, 6 ]
[ 3, 4 ]
[ 3, 5 ]
[ 3, 6 ]

7.2-2 UnorderedPairsIterator
‣ UnorderedPairsIterator( iter )( operation )

This operation returns pairs [x,y] where x,y are output from a given iterator iter. Unlike the output from CartesianIterator(iter,iter), unordered pairs are returned. In the case L = [1,2,3,...] the pairs are ordered as [1,1],[1,2],[2,2],[1,3],[2,3],[3,3],....


gap> L := [6,7,8,9];;
gap> iterL := IteratorList( L );; 
gap> pairsL := UnorderedPairsIterator( iterL );;                              
gap> while not IsDoneIterator(pairsL) do Print(NextIterator(pairsL),"\n"); od;
[ 6, 6 ]
[ 6, 7 ]
[ 7, 7 ]
[ 6, 8 ]
[ 7, 8 ]
[ 8, 8 ]
[ 6, 9 ]
[ 7, 9 ]
[ 8, 9 ]
[ 9, 9 ]
gap> iter4 := IteratorList( [ 4 ] );
<iterator>
gap> pairs4 := UnorderedPairsIterator(iter4);
<iterator>
gap> NextIterator( pairs4 );
[ 4, 4 ]
gap> IsDoneIterator( pairs4 );
true
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ՐKla mߟB&Rn@^n B&uѺ(//P6U%QٖD`ӏǥF&[3[D8EeOX{IA+Ua+$TW:ZBWKR0涀CFͅ[ Ȥ$aòQhd%|d./].侾A*%muΩu%"F\t8-B9pѴps@Fc!$sk!~)LD<~e_1̎y45ҮvY$X)~KVrdȄ9'Ar^ZZf6(n= >I#nN@_m e)cZ"W|9UtiXwc \FfJXkDrDLӉ^EER?+c'gFyC_ @}αjLn)Pt/ */ߡQz",x˶;Cˬѽ1ZpY;#V~&~8EFhTrU4w{(XA~b*'{n7 i",Fm!ݩ#jva#ܬD@΍\1A\ˆCOV= 4fBW-ƍyeu|3b("X&{Wb,l XaW_M'mvӤ2z:c=]ʸGJrZB!y7sƝ:`G)nB>Kߪbt* Tv7iMW]7maHpt}U!|+'q>#3ɿR<}!֫T,CB}?da .iPz b4Fr-X i u73Ct'P!!+Elk+}s¤{.C=i&mw5h|LZ `DJfYGIr ^.kin+&GU;S Bk~?oKf{dⴲή·~+4dK#y9~|vJaٽ.aژ :,E[׏*D((,y Cll bDyjtg9"a6 lAa&rC;:t^xr hAē~tmlRRVloEvCc%~/ɖ $[ș$%5 $j/͋̓k9#Z@lka or6V(Y)" vh 'T~c+_wf }<&-`P-a7Vk#qJVC>T`r00O `t= ң bSwz@anP;̇`قM h#9 xXL޻M)[\qy'9cR⟶C*Vh=b`vlR3k/µxWXk"‹'2^ɨM-3ėWw4z˔Tʏ=Rm*xyFe5LVce-HШ3<³c ;v }z {辫;iŔAhMQ7tH"=vk"{ҚqEjVCy)Z 4> stream xmTMo0Wx$ ! 8l[jWHL7IPV=M̼ su;Uٛ=w]yil;<[[j<=?׾+v`&ߴț<^*;~&Q>MS >_P{=s@dkx;`VY`s4JaQܡn.Uu9\Y6><ٴ.Z.4>Dӗ}~r:-d0VWk,8yLһʮӮђ[*mLr?q 5F8@=@)& 8Rx uD\j2HV0CzL] bctI g$`htы0\F0s jd< I6zg W qȐ+#k .bsrbmXK7ǵH7Gnb>&jؐu1VljOu$՟qWS/%1{\xB!K(hHTЖ枃Jρϯv=k2UKς_:~$/ ~E+7ˢ/ l(/} -+ZXukoԝE?ZKq endstream endobj 701 0 obj << /Length 739 /Filter /FlateDecode >> stream xmUMo0WxvHUdCmU^!1H#x?gx]OTm$|͜s_Iss :L;<Sz==׾f`*_`ɫڟk3'iѴ}=M;7rfnj-eSӵOLg~8 )ok A8 $`I\3`Af<Z]! xNky"7 _㓧q H`nḱRONH=CpB:# =%888QA~!*zƜАT?!~> tw8y*sύ }nFE>7*QύR>7G];~<6OIyktg>O:yұϓN|I/|yIg>O:y҅ϓ.}2 L> stream xmUMo0WxvHB!qض*jn$HP#x?gxLT$|+$=wwY[L5Okˍ}M=ٝP7{=,yfܢ_ybsn yS6`z¦}TEA] $rwyś~0uoMd?tNC0}*f6` `bێh[W0ƂtmeӶ4ݶ0[*0M\B+vX*+T*Xb-L s[ #*X,caq\``2Iш P]QA2E;XXJKC k88pLB$qƩ/088?rxy!B=X y82VAנp"Zqx8t9MD/W)u8|}ۆ~)30|SRHCOt$"NN_h 1'>4$OOB:]*N:qJ(sB5Qύ}nTsύ(Q<6O͓yk'<ϓ|t'=y '|yҩϓ|t <)>Ozg~x^uEtϭ{ՍȧU" t endstream endobj 703 0 obj << /Length 740 /Filter /FlateDecode >> stream xmUMo0WxvHB!qض*jn$HP#x?gxLT$|+$=wwY[L5Okˍ}M=4$OOB:]*N:qJ(sB5Qύ}nTsύ(Q<6O͓yk'<ϓ|t'=y '|yҩϓ|t <)>Ozg~x^uEtϭ{ՍȧU"  endstream endobj 704 0 obj << /Length 900 /Filter /FlateDecode >> stream xmUMo:W5?$R. d9M eCkmCp;;w~>|3E_?O]5߶w]Occ]=~?}Oyh9%?۹׬B|Ɯ>);vw%g43>\ 6 EJ78 1{~`W(-;]%=xe_,b+-O;q\L}UI--=BKE1p[! Mߊyu>.N5K)Wb٬8i[_uʕMzQ)V(Txޢjy!Z2P="Zd0\ÃGR\).2*Шa!U,H`+j.5Nα@VK-x%3%AYӀzΚ>kP#5m0Woþj.ZT$X/)n)#Wo(oRZ $Kp4Z-b\1ܰJ P"GXQi/8k^Zq:Zs9dB )sL-7xJ`aɽ)f$1 dъcCZC<73JgznHȰYɚTa,_-O87}KԴܗLloK+gJ.GZyVc48Wt]:P~`rZq.n1] S/Pu7Ue:?&?!d&1yHn5)yғBx#1ޞ]Go׏M?X endstream endobj 705 0 obj << /Length 851 /Filter /FlateDecode >> stream x}Un0+CW`$ MEDbɐCuA6ّD^}{l;~\N=Cu.7զOu½t}5sM7dz0;nMfMK-ٖ?xN[E_EzIVCH޾&w!X.ouc׿z|<:Q:XTߙOoGBZ ;ww ZԲWcv+>7+>S} ~!>_Sϔ+>cB|&LOr`B,&+jwRP{xᇣI^U E'b\!'}sG =C:wWq8"~t]Q|" endstream endobj 706 0 obj << /Length 851 /Filter /FlateDecode >> stream x}Un0+CW`$ MEDbɐCuA6ّD^}{l?YtգиY}w 9]7puewSClݤMӍ'oܺ sR^}5s8dz8'7&3Hy:LK-ٖ?0N[l "N/rDz=49z}'wt>X.oFzoкнz|::Q:XTߙOo'BZO KWjY׫u{FRsuB`ilB =@ )U 9yI(ѥ S*043``]߻A 𭑦 btX& #q,pg'~ι58|%Nb'QDa 8g"h~ ' ~Xkzǚ_81Bx.BzҬÀhƘ'g 2xk=6u2,bق6E0F,eL燆LY` YecODV3Μ蛳;zr֟P.O0{S3ux9(uF: }6,V|ƌV|gegV|F_+>O+>G|V|~+>C1 V|B|FB|/g)g1{!>_|&~'a9i0K!cB{XTK5;)NŽbPq<${y儢 19螡SsWѩ?!~t]QiT endstream endobj 707 0 obj << /Length 851 /Filter /FlateDecode >> stream x}Un0+CW`$ MEDbɐCUA6ّD^}{l?Ytգ;q\]U}s9n\i|ٺQ]jOtusR^}5s89Ηfi<WOlKQn8N[-#;skQ70j(+o!$oΣ;n},j7Fzoкнzr::Q:XTߙOo'BZ;vv݋ ZԲW<'suB`ilB =@ )U 9yI(ѥ S*043``MSin|kiCXc, pDˆzA:x0)ljsn l9u}SrI4"nXCA8%&ٵ6AIǚc:7^EHOupQF^odž1BЖEQ?[0^׆ƨАԗ0 9+ãbLi~jЙ}s~zrCOe fYJ|֟uМ8gΈrYφ}ŊϘъ1LҊkgigϘ݊og3f3|3ߊY[3 =L3f/gd ,' f)Rx jb&'W *~8d0UPt" ~7a3t> stream x}Un0+CW`$ MEDbɐCUA6ّD^}{l?Ytգ;q\]U}s9n\i|ٺQ]jOtusR^}5s8dz8;_Nnft_= [-E~|;en>ڲ?bs0qj>vzЌBRe]*q8cMC0c F F 14Ej~1F60aG+ıcW c rn q9Qܗ8% DMq.5Sh]`4$a]~9Vk?qc.\Y7ь1O*ezl,d mY50ymȋ,aYʘ8 xA} /Y1<*T71މf 97g19w(g1?\֟`g Yg 9LsQ.(ulgYˊx/V|V|&٭ V|N+>cv+>7+>S} ~!>_Sϔ+>cB|&LOr`B,&+jwRP{xᇣI^U E'b\!'}s =C:CޯSBQY endstream endobj 709 0 obj << /Length 851 /Filter /FlateDecode >> stream x}Un0+CW`$ MEDbɐCUA6ّD^}{l?Ytգ;q\]U}s9n\i|ٺQ]jOtusR^}5s8PYs/'7F3Hy:K-ٖ?p>ݝ2ZkXwm1985];B U{hFYs!yvq`TGy7{ E]&AjZu/?vG_L|z;9ڰƦokܰ^\zu \g$y?qמk(_KdR$| 4hd52HHNsL F.8XV`TR!fn"_LS5w[#McL#F X1+N978Nsk`q KpN8q )q4ϮEp O.?5Yş81Bx.BzҬÀhƘ'g 2xk=6u2,bق6E0F,eL燆LY` YecODV3Μ蛳;zr֟P.O0{S3ux9(uF: }6,V|ƌV|gegV|F_+>O+>G|V|~+>C1 V|B|FB|/g)g1{!>_|&~'a9i0K!cB{XTK5;)NŽbPq<${y儢 1 螡SsWѩ?!~tM(Vu endstream endobj 710 0 obj << /Length 851 /Filter /FlateDecode >> stream x}Un0+CW`$ MEDbɐCUA6ّD^}{l?Ytգ;q\]U}s9n\i|ٺQ]jOtusR^}5s8BϚ|9a6Fa|/Xo϶Zk\ĩ$wAg LC3ʊw۷莛n˥?8ޛ`~?n8t/3NbV+պ};:5bwɩֆ56}Χ]]₥+U=#)\]PG*X$D F @F@k} 89@FJuFF#`R0Jt 5 5553G|F/+>㽴3Z~Z83f3[:٭ ߬Lg3t33 ~!>CO!>S 33>IY ?BXIAup*Çq&#{U-'H8qe%@ 8{N Y;lF?h; endstream endobj 711 0 obj << /Length 851 /Filter /FlateDecode >> stream x}Un0CƆ"Rjn"BD73o`_|nxZNcg!w㍺O겺otȷ}z9/>vⅵv?;~Mf^ [-E~e!jcc4qj~( 3F&VݮdE>8ǷvK5i|#Wn_gt8zQ:ZT緡nfߙOobZ?67ZԲiV^:} ?L; N|N|oN|3;:K}KR|)W|L|Or`R,&+jwRR{xI^ E'b\!'}{p =C:]߯p@;_X5w| endstream endobj 712 0 obj << /Length 1022 /Filter /FlateDecode >> stream xmVMo8Wh҃kTHrضh^I IJ!ۇf|tǙqV}xܟ>ڿ7]Ocp{Vc> stream xmTMk0WhFG*! miʲVZCcYy#9톅ļ{3񼛤es^7箰 nn8l=hzI-._뫦~^JIu]f `tTsr*o8{&X,dew+mWos~X(2X.EiTz}ܟ^7uY~lVNMєo R.bY.֔O9؄b%9vsr(MXa#D$ar bqMDs!FKRLDP0.BEHQ#͸FuŎ577v}QȕanOd$g;A,əCR;6+ѧx**Ę$90q'oקfQ%n;5pX2]$^q~+s"F!CyhIh~CMnOf1$#h)r~hмj5F̹k#ni<7>Tsa>s\8s&wsaY1:+r1\ut[ZM,k4w6_%aJ endstream endobj 714 0 obj << /Length 665 /Filter /FlateDecode >> stream xmTMk0WhFG*! miʲVZCcYy#9햅ļ{3񸟤e&Oo]&C]]Mq>zwt߉Ǯ)n.pCx?nڽVgx=itO"i [\l\WM}'ԭ̚t4pXeȉeU oq yM\-CnCW_Ey}wP dZz891euB)] W-\v\]~[S!8&+Zce"'2Ɍ5I@|"B2AQhSlLء28a}ɑFq5ҍnnbfǮCG= Wܢe$g;A,:sx l=NOTƘ$0_س/vЧQ%~Zx pX2]$^qnaK??q FqMyc0=) &l(mi,3|d &\c ]͹&ӈ9w{d-tx\ \cΜekqLJs?<@>qhx .׷8wl~1V<*m"mmDa endstream endobj 715 0 obj << /Length 990 /Filter /FlateDecode >> stream xmVnH+AMyثL,R~(r]]n7W~c7߽p}z\[{%8OeWϝ\'}oe<]8m|kq7,r[|UJ;P)ڮ:U8 *&~(Ia7~x;~z. 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Operations from AutoDoc The file functions FindMatchingFiles FindMatchingFiles and CreateDirIfMissing CreateDirIfMissing were copied from package AutoDoc where they are named AutoDoc_FindMatchingFiles and AutoDoc_CreateDirIfMissing.

The string function StringDotSuffix StringDotSuffix was also copied from package AutoDoc, where it is named AUTODOC_GetSuffix. GetSuffix

The function SetIfMissing SetIfMissing was also transferred from package AutoDoc, where it is called AUTODOC_SetIfMissing. It writes into a record provided the position is not yet bound.

As from version 0.61, all these functions became obsolete in Utils, but continue to be defined in AutoDoc.

Functions for printing The function PrintOneItemPerLine PrintOneItemPerLine was used to prints lists vertically, rather than horizontally. Since a very similar result may be achieved using the &GAP; library functions Perform and Display, this function became obsolete in version 0.61.

s3 := SymmetricGroup( 3 );; gap> L := KnownPropertiesOfObject( GeneratorsOfGroup( s3 ) );; gap> Perform( L, Display ); IsFinite IsSmallList IsGeneratorsOfMagmaWithInverses IsGeneratorsOfSemigroup IsSubsetLocallyFiniteGroup gap> Perform( s3, Display ); () (2,3) (1,3) (1,3,2) (1,2,3) (1,2) ]]>

Other obsolete functions Applicable Methods The function PrintApplicableMethod PrintApplicableMethod, which was included in versions from 0.41 to 0.58, has been removed since it was considered superfluous. The example shows how to print out a function.

ApplicableMethod( IsCyclic, [ Group((1,2,3),(4,5)) ], 1, 1 ); #I Searching Method for IsCyclic with 1 arguments: #I Total: 7 entries #I Method 4: ``IsCyclic'' at /Applications/gap/gap4r9/lib/grp.gi:30 , value: 36 function( G ) ... end gap> Print( last ); function ( G ) if Length( GeneratorsOfGroup( G ) ) = 1 then return true; else TryNextMethod(); fi; return; end gap> ApplicableMethod( IsCyclic, [ Group((1,2,3),(4,5)) ], 0, 3 ); function( <1 unnamed arguments> ) ... end gap> Print( last ); function ( <> ) <> end ]]> ExponentOfPrime The function ExponentOfPrime ExponentOfPrime was originally transferred from package &RCWA;. The command ExponentOfPrime(n,p) returned the exponent of the prime p in the prime factorization of n.

Since the &GAP; function PValuation produces the same results, and does so more quickly, this function has been made obsolete.

utils-0.86/doc/times.css000644 000765 000024 00000000261 14621610515 020313 0ustar00christopherwensleystaff000000 000000 /* times.css Frank Lübeck */ /* Change default CSS to use Times font. */ body { font-family: Times,Times New Roman,serif; } utils-0.86/doc/utils.xml000644 000765 000024 00000001257 14621610512 020345 0ustar00christopherwensleystaff000000 000000 ] > <#Include SYSTEM "title.xml"> <#Include SYSTEM "intro.xml"> <#Include SYSTEM "print.xml"> <#Include SYSTEM "lists.xml"> <#Include SYSTEM "number.xml"> <#Include SYSTEM "groups.xml"> <#Include SYSTEM "matrix.xml"> <#Include SYSTEM "iterator.xml"> <#Include SYSTEM "record.xml"> <#Include SYSTEM "download.xml"> <#Include SYSTEM "others.xml"> <#Include SYSTEM "obsolete.xml"> <#Include SYSTEM "transfer.xml"> utils-0.86/doc/toggless.css000644 000765 000024 00000001672 14621610515 021030 0ustar00christopherwensleystaff000000 000000 /* toggless.css Frank Lübeck */ /* Using javascript we change all div.ContSect to div.ContSectOpen or div.ContSectClosed. This way the config for div.ContSect in manual.css is no longer relevant. Here we add the CSS for the new elements. */ /* This layout is based on an idea by Burkhard Höfling. */ div.ContSectClosed { text-align: left; margin-left: 1em; } div.ContSectOpen { text-align: left; margin-left: 1em; } div.ContSectOpen div.ContSSBlock { display: block; text-align: left; margin-left: 1em; } div.ContSectOpen div.ContSSBlock a { display: block; width: 100%; margin-left: 1em; } span.tocline a:hover { display: inline; background: #eeeeee; } span.ContSS a:hover { display: inline; background: #eeeeee; } span.toctoggle { font-size: 80%; display: inline-block; width: 1.2em; } span.toctoggle:hover { background-color: #aaaaaa; } utils-0.86/doc/chap2.txt000644 000765 000024 00000005356 14621610512 020225 0ustar00christopherwensleystaff000000 000000 2 Printing Lists and Iterators 2.1 Printing selected items The functions described here print lists or objects with an iterator with one item per line, either the whole list/iterator or certain subsets:  by giving a list of positions of items to be printed, or  by specifying a first item and then a regular step. 2.1-1 PrintSelection PrintSelection( obj, first, step[, last] )  function PrintSelection( obj, list )  function This function, given three (or four) parameters, calls operations PrintSelectionFromList or PrintSelectionFromIterator which prints the first item specified, and then the item at every step. The fourth parameter is essential when the object being printed is infinite. Alternatively, given two parameters, with the second parameter a list L of positive integers, only the items at positions in L are printed.  Example   gap> L := List( [1..20], n -> n^5 );; gap> PrintSelection( L, [18..20] ); 18 : 1889568 19 : 2476099 20 : 3200000 gap> PrintSelection( L, 2, 9 );  2 : 32 11 : 161051 20 : 3200000 gap> PrintSelection( L, 2, 3, 11 ); 2 : 32 5 : 3125 8 : 32768 11 : 161051 gap> s5 := SymmetricGroup( 5 );; gap> PrintSelection( s5, [30,31,100,101] ); 30 : (1,5)(3,4) 31 : (1,5,2) 100 : (1,4,3) 101 : (1,4)(3,5) gap> PrintSelection( s5, 1, 30 ); 1 : () 31 : (1,5,2) 61 : (1,2,3) 91 : (1,3,5,2,4) gap> PrintSelection( s5, 9, 11, 43 ); 9 : (2,5,3) 20 : (2,4) 31 : (1,5,2) 42 : (1,5,2,3,4)   utils-0.86/doc/chap3.txt000644 000765 000024 00000024016 14621610512 020220 0ustar00christopherwensleystaff000000 000000 3 Lists, Sets and Strings 3.1 Functions for lists 3.1-1 DifferencesList DifferencesList( L )  function This function has been transferred from package ResClasses. It takes a list L of length n and outputs the list of length n-1 containing all the differences L[i]-L[i-1].  Example   gap> List( [1..12], n->n^3 ); [ 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 ] gap> DifferencesList( last ); [ 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397 ] gap> DifferencesList( last ); [ 12, 18, 24, 30, 36, 42, 48, 54, 60, 66 ] gap> DifferencesList( last ); [ 6, 6, 6, 6, 6, 6, 6, 6, 6 ]   3.1-2 QuotientsList QuotientsList( L )  function FloatQuotientsList( L )  function These functions have been transferred from package ResClasses. They take a list L of length n and output the quotients L[i]/L[i-1] of consecutive entries in L. An error is returned if an entry is zero.  Example   gap> List( [0..10], n -> Factorial(n) ); [ 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 ] gap> QuotientsList( last ); [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ] gap> L := [ 1, 3, 5, -1, -3, -5 ];; gap> QuotientsList( L ); [ 3, 5/3, -1/5, 3, 5/3 ] gap> FloatQuotientsList( L ); [ 3., 1.66667, -0.2, 3., 1.66667 ] gap> QuotientsList( [ 2, 1, 0, -1, -2 ] ); [ 1/2, 0, fail, 2 ] gap> FloatQuotientsList( [1..10] ); [ 2., 1.5, 1.33333, 1.25, 1.2, 1.16667, 1.14286, 1.125, 1.11111 ] gap> Product( last ); 10.    3.1-3 SearchCycle SearchCycle( L )  operation This function has been transferred from package RCWA. SearchCycle is a tool to find likely cycles in lists. What, precisely, a cycle is, is deliberately fuzzy here, and may possibly even change. The idea is that the beginning of the list may be anything, following that the same pattern needs to be repeated several times in order to be recognized as a cycle.  Example   gap> L := [1..20];; L[1]:=13;;  gap> for i in [1..19] do  >  if IsOddInt(L[i]) then L[i+1]:=3*L[i]+1; else L[i+1]:=L[i]/2; fi; >  od;  gap> L;  [ 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4, 2, 1, 4 ] gap> SearchCycle( L );  [ 1, 4, 2 ] gap> n := 1;; L := [n];; gap> for i in [1..100] do n:=(n^2+1) mod 1093; Add(L,n); od; gap> L;  [ 1, 2, 5, 26, 677, 363, 610, 481, 739, 715, 795, 272, 754, 157, 604, 848,   1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271,   211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521,   378, 795, 272, 754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272,   754, 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604,   848, 1004, 271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004,   271, 211, 802, 521, 378, 795, 272, 754, 157, 604, 848, 1004 ] gap> C := SearchCycle( L ); [ 157, 604, 848, 1004, 271, 211, 802, 521, 378, 795, 272, 754 ] gap> P := Positions( L, 157 ); [ 14, 26, 38, 50, 62, 74, 86, 98 ] gap> Length( C ); DifferencesList( P ); 12 [ 12, 12, 12, 12, 12, 12, 12 ]   3.1-4 RandomCombination RandomCombination( S, k )  operation This function has been transferred from package ResClasses. It returns a random unordered k-tuple of distinct elements of a set S.  Example   gap> ## "6 aus 49" is a common lottery in Germany gap> RandomCombination( [1..49], 6 );  [ 2, 16, 24, 26, 37, 47 ]   3.2 Distinct and Common Representatives 3.2-1 DistinctRepresentatives DistinctRepresentatives( list )  operation CommonRepresentatives( list )  operation CommonTransversal( grp, subgrp )  operation IsCommonTransversal( grp, subgrp, list )  operation These operations have been transferred from package XMod. They deal with lists of subsets of [1 ... n] and construct systems of distinct and common representatives using simple, non-recursive, combinatorial algorithms. When L is a set of n subsets of [1 ... n] and the Hall condition is satisfied (the union of any k subsets has at least k elements), a set of DistinctRepresentatives exists. When J,K are both lists of n sets, the operation CommonRepresentatives returns two lists: the set of representatives, and a permutation of the subsets of the second list. The operation CommonTransversal may be used to provide a common transversal for the sets of left and right cosets of a subgroup H of a group G, although a greedy algorithm is usually quicker.  Example   gap> J := [ [1,2,3], [3,4], [3,4], [1,2,4] ];; gap> DistinctRepresentatives( J ); [ 1, 3, 4, 2 ] gap> K := [ [3,4], [1,2], [2,3], [2,3,4] ];; gap> CommonRepresentatives( J, K ); [ [ 3, 3, 3, 1 ], [ 1, 3, 4, 2 ] ] gap> d16 := DihedralGroup( IsPermGroup, 16 );  Group([ (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) ]) gap> SetName( d16, "d16" ); gap> c4 := Subgroup( d16, [ d16.1^2 ] );  Group([ (1,3,5,7)(2,4,6,8) ]) gap> SetName( c4, "c4" ); gap> RightCosets( d16, c4 ); [ RightCoset(c4,()), RightCoset(c4,(2,8)(3,7)(4,6)), RightCoset(c4,(1,8,7,6,5,  4,3,2)), RightCoset(c4,(1,8)(2,7)(3,6)(4,5)) ] gap> trans := CommonTransversal( d16, c4 ); [ (), (2,8)(3,7)(4,6), (1,2,3,4,5,6,7,8), (1,2)(3,8)(4,7)(5,6) ] gap> IsCommonTransversal( d16, c4, trans ); true   3.3 Functions for strings 3.3-1 BlankFreeString BlankFreeString( obj )  function This function has been transferred from package ResClasses. The result of BlankFreeString( obj ); is a composite of the functions String( obj ) and RemoveCharacters( obj, " " );.  Example   gap> gens := GeneratorsOfGroup( DihedralGroup(12) ); [ f1, f2, f3 ] gap> String( gens );  "[ f1, f2, f3 ]" gap> BlankFreeString( gens );  "[f1,f2,f3]"   utils-0.86/doc/chapBib.html000644 000765 000024 00000013540 14621610515 020702 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - References
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References

[BCP97] Bosma, W., Cannon, J. and Playoust, C., The Magma algebra system. {I}. The user language, J. Symbolic Comput., 24 (3-4) (1997), 235--265 pages
( Computational algebra and number theory (London, 1993)} https://doi.org/10.1006/jsco.1996.0125 ).

[GH16] Gutsche, S. and Horn, M., AutoDoc - Generate documentation from GAP source code (Version 2016.12.04) (2016)
( GAP package, https://github.com/gap-packages/AutoDoc ).

[Hor17] Horn, M., GitHubPagesForGAP - Template for easily using GitHub Pages within GAP packages (Version 0.2) (2017)
( GAP package, https://gap-system.github.io/GitHubPagesForGAP/ ).

[Koh17a] Kohl, S., RCWA - Residue-Class-Wise Affine Groups (Version 4.5.1) (2017)
( GAP package, https://stefan-kohl.github.io/rcwa.html ).

[Koh17b] Kohl, S., ResClasses - Set-Theoretic Computations with Residue Classes (Version 4.6.0) (2017)
( GAP package, https://stefan-kohl.github.io/resclasses.html ).

[LN17] Lübeck, F. and Neunhöffer, M., GAPDoc (Version 1.6), RWTH Aachen (2017)
( GAP package, https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/index.html ).

[WAOU17] Wensley, C. D., Alp, M., Odabas, A. and Uslu, E. O., XMod - Crossed Modules and Cat1-groups in GAP (Version 2.64) (2017)
( GAP package, https://github.com/gap-packages/xmod ).

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10 Various other functions
 10.1 File operations

  10.1-1 Log2HTML
 10.2 LaTeX strings

  10.2-1 IntOrOnfinityToLaTeX
  10.2-2 LaTeXStringFactorsInt
 10.3 Conversion to \({\sf Magma}\) string

  10.3-1 ConvertToMagmaInputString

10 Various other functions

10.1 File operations

10.1-1 Log2HTML
‣ Log2HTML( filename )( function )

This function has been transferred from package RCWA.

This function converts the GAP logfile filename to HTML. It appears that the logfile should be in your current directory. The extension of the input file must be *.log. The name of the output file is the same as the one of the input file except that the extension *.log is replaced by *.html. There is a sample CSS file in utils/doc/gaplog.css, which you can adjust to your taste.


gap> LogTo( "triv.log" );
gap> a := 33^5;
39135393
gap> LogTo(); 
gap> Log2HTML( "triv.log" );

10.2 LaTeX strings

10.2-1 IntOrOnfinityToLaTeX
‣ IntOrOnfinityToLaTeX( n )( function )

This function has been transferred from package ResClasses.

IntOrInfinityToLaTeX(n) returns the LaTeX string for n.


gap> IntOrInfinityToLaTeX( 10^3 );
"1000"
gap> IntOrInfinityToLaTeX( infinity );
"\\infty"

10.2-2 LaTeXStringFactorsInt
‣ LaTeXStringFactorsInt( n )( function )

This function has been transferred from package RCWA.

It returns the prime factorization of the integer \(n\) as a string in LaTeX format.


gap> LaTeXStringFactorsInt( Factorial(12) );
"2^{10} \\cdot 3^5 \\cdot 5^2 \\cdot 7 \\cdot 11"

10.3 Conversion to \({\sf Magma}\) string

10.3-1 ConvertToMagmaInputString
‣ ConvertToMagmaInputString( arg )( function )

The function ConvertToMagmaInputString( obj [, str] ) attempts to output a string s which can be read into \({\sf Magma}\) [BCP97] so as to produce the same group in that computer algebra system. In the second form the user specifies the name of the resulting object, so that the output string has the form "str := ...". When obj is a permutation group, the operation PermGroupToMagmaFormat(obj) is called. This function has been taken from other.gi in the main library where it was called MagmaInputString. When obj is a pc-group, the operation PcGroupToMagmaFormat(obj) is called. This function was private code of Max Horn. When obj is a matrix group over a finite field, the operation MatrixGroupToMagmaFormat(obj) is called. This function is a modification of private code of Frank Lübeck.

Hopefully code for other types of group will be added in due course.

These functions should be considered experimental, and more testing is desirable.


gap> ConvertToMagmaInputString( Group( (1,2,3,4,5), (3,4,5) ) );
"PermutationGroup<5|(1,2,3,4,5),\n(3,4,5)>;\n"
gap> ConvertToMagmaInputString( Group( (1,2,3,4,5) ), "c5" );        
"c5:=PermutationGroup<5|(1,2,3,4,5)>;\n"
gap> ConvertToMagmaInputString( DihedralGroup( IsPcGroup, 10 ) );
"PolycyclicGroup< f1,f2 |\nf1^2,\nf2^5,\nf2^f1 = f2^4\n>;\n"
gap> M := GL(2,5);;  Size(M); 
480
gap> s1 := ConvertToMagmaInputString( M );
"F := GF(5);\nP := GL(2,F);\ngens := [\nP![2,0,0,1],\nP![4,1,4,0]\n];\nsub<P |\
 gens>;\n"
gap> Print( s1 );
F := GF(5);
P := GL(2,F);
gens := [
P![2,0,0,1],
P![4,1,4,0]
];
sub<P | gens>;
gap> n1 := [ [ Z(9)^0, Z(9)^0 ], [ Z(9)^0, Z(9) ] ];;
gap> n2 := [ [ Z(9)^0, Z(9)^3 ], [ Z(9)^4, Z(9)^2 ] ];;
gap> N := Group( n1, n2 );;  Size( N );
5760
gap> s2 := ConvertToMagmaInputString( N, "gpN" );;
gap> Print( s2 );
F := GF(3^2);
P := GL(2,F);
w := PrimitiveElement(F);
gens := [
P![ 1, 1, 1,w^1],
P![ 1,w^3, 2,w^2]
];
gpN := sub<P | gens>;
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Functions for integers These functions are in the process of being transferred from package &Home;.

They

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utils

Utility functions in GAP

0.86

16 May 2024

Thomas Breuer
Email: sam@math.rwth-aachen.de
Homepage: https://www.math.rwth-aachen.de/~Thomas.Breuer

Sebastian Gutsche
Email: gutsche@mathematik.uni-siegen.de
Homepage: https://sebasguts.github.io/

Max Horn
Email: mhorn@rptu.de
Homepage: https://github.com/mhorn

Alexander Hulpke
Email: hulpke@math.colostate.edu
Homepage: https://www.math.colostate.edu/~hulpke

Pedro García-Sánchez
Email: pedro@ugr.es
Homepage: http://www.ugr.es/local/pedro

Christopher Jefferson
Email: caj21@st-andrews.ac.uk
Homepage: https://caj.host.cs.st-andrews.ac.uk/

Stefan Kohl
Email: stefan@mcs.st-and.ac.uk
Homepage: https://www.gap-system.org/DevelopersPages/StefanKohl/

Frank Lübeck
Email: Frank.Luebeck@Math.RWTH-Aachen.De
Homepage: https://www.math.rwth-aachen.de/~Frank.Luebeck

Chris Wensley
Email: cdwensley.maths@btinternet.com
Homepage: https://github.com/cdwensley

Abstract

The Utils package provides a space for utility functions in a variety of GAP packages to be collected together into a single package. In this way it is hoped that they will become more visible to package authors.

Any package author who transfers a function to Utils will become an author of Utils.

If deemed appropriate, functions may also be transferred from the main library.

Bug reports, suggestions and comments are, of course, welcome. Please contact the last author at cdwensley.maths@btinternet.com or submit an issue at the GitHub repository https://github.com/gap-packages/utils/issues/.

Copyright

© 2015-2024, The GAP Group.

The Utils package is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

Acknowledgements

This documentation was prepared using the GAPDoc [LN17] and AutoDoc [GH16] packages.

The procedure used to produce new releases uses the package GitHubPagesForGAP [Hor17] and the package ReleaseTools.

Contents

1 Introduction
 1.1 Information for package authors
2 Printing Lists and Iterators
 2.1 Printing selected items

  2.1-1 PrintSelection
3 Lists, Sets and Strings
 3.1 Functions for lists

  3.1-1 DifferencesList
  3.1-2 QuotientsList
  3.1-3 SearchCycle
  3.1-4 RandomCombination
 3.2 Distinct and Common Representatives

  3.2-1 DistinctRepresentatives
 3.3 Functions for strings

  3.3-1 BlankFreeString
4 Number-theoretic functions
 4.1 Functions for integers

  4.1-1 AllSmoothIntegers
  4.1-2 AllProducts
  4.1-3 RestrictedPartitionsWithoutRepetitions
  4.1-4 NextProbablyPrimeInt
  4.1-5 PrimeNumbersIterator
5 Groups and homomorphisms
 5.1 Functions for groups

  5.1-1 Comm
  5.1-2 IsCommuting
  5.1-3 ListOfPowers
  5.1-4 GeneratorsAndInverses
  5.1-5 UpperFittingSeries
 5.2 Left Cosets for Groups

  5.2-1 LeftCoset
  5.2-2 Inverse
 5.3 Functions for group homomorphisms

  5.3-1 EpimorphismByGenerators
  5.3-2 Pullback
  5.3-3 CentralProduct
  5.3-4 IdempotentEndomorphisms
  5.3-5 DirectProductOfFunctions
  5.3-6 DirectProductOfAutomorphismGroups
6 Matrices
 6.1 Some operations for matrices

  6.1-1 DirectSumDecompositionMatrices
7 Iterators
 7.1 Some iterators for groups and their isomorphisms

  7.1-1 AllIsomorphismsIterator
  7.1-2 AllSubgroupsIterator
 7.2 Operations on iterators

  7.2-1 CartesianIterator
  7.2-2 UnorderedPairsIterator
8 Records
 8.1 Functions for records

  8.1-1 AssignGlobals
 8.2 Option records for functions

  8.2-1 OptionRecordWithDefaults
9 Web Downloads
 9.1 Functions for downloading files from the web

  9.1-1 Download
  9.1-2 User preference DownloadVerifyCertificate
10 Various other functions
 10.1 File operations

  10.1-1 Log2HTML
 10.2 LaTeX strings

  10.2-1 IntOrOnfinityToLaTeX
  10.2-2 LaTeXStringFactorsInt
 10.3 Conversion to Magma string

  10.3-1 ConvertToMagmaInputString
11 Obsolete functions
 11.1 Operations from AutoDoc
 11.2 Functions for printing
 11.3 Other obsolete functions

  11.3-1 Applicable Methods
  11.3-2 ExponentOfPrime
12 The transfer procedure
References
Index

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utils-0.86/doc/chap1.txt000644 000765 000024 00000012723 14621610512 020220 0ustar00christopherwensleystaff000000 000000 1 Introduction The Utils package provides a space for utility functions from a variety of GAP packages to be collected together into a single package. In this way it is hoped that they will become more visible to other package authors. This package was first distributed as part of the GAP 4.8.2 distribution. The package is loaded with the command  Example   gap> LoadPackage( "utils" );    Functions have been transferred from the following packages:  Conversion of a GAP group to a Magma output string, taken from various sources including other.gi in the main library. Transfer is complete (for now) for functions from the following packages:  AutoDoc [GH16] (with function names changed);  ResClasses [Koh17b];  RCWA [Koh17a];  XMod [WAOU17]. The package may be obtained either as a compressed .tar file or as a .zip file, utils-version_number.tar.gz, by ftp from one of the following sites:  the Utils GitHub release site: https://gap-packages.github.io/utils/.  any GAP archive, e.g. https://www.gap-system.org/Packages/packages.html; The package also has a GitHub repository at: https://github.com/gap-packages/utils. Once the package is loaded, the manual doc/manual.pdf can be found in the documentation folder. The html versions, with or without MathJax, may be rebuilt as follows:  Example   gap> ReadPackage( "utils", "makedoc.g" );    It is possible to check that the package has been installed correctly by running the test files (which terminates the GAP session):  Example   gap> ReadPackage( "utils", "tst/testall.g" ); Architecture: . . . . .  testing: . . . . .  . . .  #I No errors detected while testing   Note that functions listed in this manual that are currently in the process of being transferred are only read from the source package Home (say), and so can only be used if Home has already been loaded. There are no such functions in transition at present. 1.1 Information for package authors A function (or collection of functions) is suitable for transfer from a package Home to Utils if the following conditions are satisfied.  The function is sufficiently non-specialised so that it might be of use to other authors.  The function does not depend on the remaining functions in Home  The function does not do what can already be done with a GAP library function.  Documentation of the function and test examples are available.  When there is more than one active author of Home, they should all be aware (and content) that the transfer is taking place. Authors of packages may be reluctant to let go of their utility functions. The following principles may help to reassure them. (Suggestions for more items here are welcome.)  A function that has been transferred to Utils will not be changed without the approval of the original author.  The current package maintainer has every intention of continuing to maintain Utils. In the event that this proves impossible, the GAP development team will surely find someone to take over.  Function names will not be changed unless specifically requested by Home's author(s) or unless they have the form HOME_FunctionName.  In order to speed up the transfer process, only functions from one package will be in transition at any given time. Hopefully a week or two will suffice for most packages.  Any package author who transfers a function to Utils will become an author of Utils. (In truth, Utils does not have authors, just a large number of contributors.) The process for transferring utility functions from Home to Utils is described in Chapter 12. utils-0.86/doc/chap0.txt000644 000765 000024 00000015106 14621610512 020215 0ustar00christopherwensleystaff000000 000000  utils   Utility functions in GAP  0.86 16 May 2024 Thomas Breuer Sebastian Gutsche Max Horn Alexander Hulpke Pedro García-Sánchez Christopher Jefferson Stefan Kohl Frank Lübeck Chris Wensley Thomas Breuer Email: mailto:sam@math.rwth-aachen.de Homepage: https://www.math.rwth-aachen.de/~Thomas.Breuer Sebastian Gutsche Email: mailto:gutsche@mathematik.uni-siegen.de Homepage: https://sebasguts.github.io/ Max Horn Email: mailto:mhorn@rptu.de Homepage: https://github.com/mhorn Alexander Hulpke Email: mailto:hulpke@math.colostate.edu Homepage: https://www.math.colostate.edu/~hulpke Pedro García-Sánchez Email: mailto:pedro@ugr.es Homepage: http://www.ugr.es/local/pedro Christopher Jefferson Email: mailto:caj21@st-andrews.ac.uk Homepage: https://caj.host.cs.st-andrews.ac.uk/ Stefan Kohl Email: mailto:stefan@mcs.st-and.ac.uk Homepage: https://www.gap-system.org/DevelopersPages/StefanKohl/ Frank Lübeck Email: mailto:Frank.Luebeck@Math.RWTH-Aachen.De Homepage: https://www.math.rwth-aachen.de/~Frank.Luebeck Chris Wensley Email: mailto:cdwensley.maths@btinternet.com Homepage: https://github.com/cdwensley ------------------------------------------------------- Abstract The Utils package provides a space for utility functions in a variety of GAP packages to be collected together into a single package. In this way it is hoped that they will become more visible to package authors. Any package author who transfers a function to Utils will become an author of Utils. If deemed appropriate, functions may also be transferred from the main library. Bug reports, suggestions and comments are, of course, welcome. Please contact the last author at mailto:cdwensley.maths@btinternet.com or submit an issue at the GitHub repository https://github.com/gap-packages/utils/issues/. ------------------------------------------------------- Copyright © 2015-2024, The GAP Group. The Utils package is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. ------------------------------------------------------- Acknowledgements This documentation was prepared using the GAPDoc [LN17] and AutoDoc [GH16] packages. The procedure used to produce new releases uses the package GitHubPagesForGAP [Hor17] and the package ReleaseTools. ------------------------------------------------------- Contents (Utils) 1 Introduction 1.1 Information for package authors 2 Printing Lists and Iterators 2.1 Printing selected items 2.1-1 PrintSelection 3 Lists, Sets and Strings 3.1 Functions for lists 3.1-1 DifferencesList 3.1-2 QuotientsList 3.1-3 SearchCycle 3.1-4 RandomCombination 3.2 Distinct and Common Representatives 3.2-1 DistinctRepresentatives 3.3 Functions for strings 3.3-1 BlankFreeString 4 Number-theoretic functions 4.1 Functions for integers 4.1-1 AllSmoothIntegers 4.1-2 AllProducts 4.1-3 RestrictedPartitionsWithoutRepetitions 4.1-4 NextProbablyPrimeInt 4.1-5 PrimeNumbersIterator 5 Groups and homomorphisms 5.1 Functions for groups 5.1-1 Comm 5.1-2 IsCommuting 5.1-3 ListOfPowers 5.1-4 GeneratorsAndInverses 5.1-5 UpperFittingSeries 5.2 Left Cosets for Groups 5.2-1 LeftCoset 5.2-2 Inverse 5.3 Functions for group homomorphisms 5.3-1 EpimorphismByGenerators 5.3-2 Pullback 5.3-3 CentralProduct 5.3-4 IdempotentEndomorphisms 5.3-5 DirectProductOfFunctions 5.3-6 DirectProductOfAutomorphismGroups 6 Matrices 6.1 Some operations for matrices 6.1-1 DirectSumDecompositionMatrices 7 Iterators 7.1 Some iterators for groups and their isomorphisms 7.1-1 AllIsomorphismsIterator 7.1-2 AllSubgroupsIterator 7.2 Operations on iterators 7.2-1 CartesianIterator 7.2-2 UnorderedPairsIterator 8 Records 8.1 Functions for records 8.1-1 AssignGlobals 8.2 Option records for functions 8.2-1 OptionRecordWithDefaults 9 Web Downloads 9.1 Functions for downloading files from the web 9.1-1 Download 9.1-2 User preference DownloadVerifyCertificate 10 Various other functions 10.1 File operations 10.1-1 Log2HTML 10.2 LaTeX strings 10.2-1 IntOrOnfinityToLaTeX 10.2-2 LaTeXStringFactorsInt 10.3 Conversion to Magma string 10.3-1 ConvertToMagmaInputString 11 Obsolete functions 11.1 Operations from AutoDoc 11.2 Functions for printing 11.3 Other obsolete functions 11.3-1 Applicable Methods 11.3-2 ExponentOfPrime 12 The transfer procedure  utils-0.86/doc/chap4_mj.html000644 000765 000024 00000030633 14621610515 021041 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Chapter 4: Number-theoretic functions
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4 Number-theoretic functions
 4.1 Functions for integers

  4.1-1 AllSmoothIntegers
  4.1-2 AllProducts
  4.1-3 RestrictedPartitionsWithoutRepetitions
  4.1-4 NextProbablyPrimeInt
  4.1-5 PrimeNumbersIterator

4 Number-theoretic functions

4.1 Functions for integers

4.1-1 AllSmoothIntegers
‣ AllSmoothIntegers( maxp, maxn )( function )
‣ AllSmoothIntegers( L, maxp )( function )

This function has been transferred from package RCWA.

The function AllSmoothIntegers(maxp,maxn) returns the list of all positive integers less than or equal to maxn whose prime factors are all in the list \(L = \{p ~|~ p \leqslant maxp, p~\mbox{prime} \}\).

In the alternative form, when \(L\) is a list of primes, the function returns the list of all positive integers whose prime factors lie in \(L\).


gap> AllSmoothIntegers( 3, 1000 );
[ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 
  108, 128, 144, 162, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576, 
  648, 729, 768, 864, 972 ]
gap> AllSmoothIntegers( [5,11,17], 1000 );
[ 1, 5, 11, 17, 25, 55, 85, 121, 125, 187, 275, 289, 425, 605, 625, 935 ]
gap> Length( last );
16
gap> List( [3..20], n -> Length( AllSmoothIntegers( [5,11,17], 10^n ) ) );
[ 16, 29, 50, 78, 114, 155, 212, 282, 359, 452, 565, 691, 831, 992, 1173, 
  1374, 1595, 1843 ]

4.1-2 AllProducts
‣ AllProducts( L, k )( function )

This function has been transferred from package RCWA.

The command AllProducts(L,k) returns the list of all products of k entries of the list L. Note that every ordering of the entries is used so that, in the commuting case, there are bound to be repetitions.


gap> AllProducts([1..4],3); 
[ 1, 2, 3, 4, 2, 4, 6, 8, 3, 6, 9, 12, 4, 8, 12, 16, 2, 4, 6, 8, 4, 8, 12, 
  16, 6, 12, 18, 24, 8, 16, 24, 32, 3, 6, 9, 12, 6, 12, 18, 24, 9, 18, 27, 
  36, 12, 24, 36, 48, 4, 8, 12, 16, 8, 16, 24, 32, 12, 24, 36, 48, 16, 32, 
  48, 64 ]
gap> Set(last);            
[ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 64 ]
gap> AllProducts( [(1,2,3),(2,3,4)], 2 );
[ (2,4,3), (1,2)(3,4), (1,3)(2,4), (1,3,2) ]

4.1-3 RestrictedPartitionsWithoutRepetitions
‣ RestrictedPartitionsWithoutRepetitions( n, S )( function )

This function has been transferred from package RCWA.

For a positive integer n and a set of positive integers S, this function returns the list of partitions of n into distinct elements of S. Unlike RestrictedPartitions, no repetitions are allowed.


gap> RestrictedPartitions( 20, [4..10] );
[ [ 4, 4, 4, 4, 4 ], [ 5, 5, 5, 5 ], [ 6, 5, 5, 4 ], [ 6, 6, 4, 4 ], 
  [ 7, 5, 4, 4 ], [ 7, 7, 6 ], [ 8, 4, 4, 4 ], [ 8, 6, 6 ], [ 8, 7, 5 ], 
  [ 8, 8, 4 ], [ 9, 6, 5 ], [ 9, 7, 4 ], [ 10, 5, 5 ], [ 10, 6, 4 ], 
  [ 10, 10 ] ]
gap> RestrictedPartitionsWithoutRepetitions( 20, [4..10] );
[ [ 10, 6, 4 ], [ 9, 7, 4 ], [ 9, 6, 5 ], [ 8, 7, 5 ] ]
gap> RestrictedPartitionsWithoutRepetitions( 10^2, List([1..10], n->n^2 ) );
[ [ 100 ], [ 64, 36 ], [ 49, 25, 16, 9, 1 ] ]

4.1-4 NextProbablyPrimeInt
‣ NextProbablyPrimeInt( n )( function )

This function has been transferred from package RCWA.

The function NextProbablyPrimeInt(n) does the same as NextPrimeInt(n) except that for reasons of performance it tests numbers only for IsProbablyPrimeInt(n) instead of IsPrimeInt(n). For large n, this function is much faster than NextPrimeInt(n)


gap> n := 2^251;
3618502788666131106986593281521497120414687020801267626233049500247285301248
gap> NextProbablyPrimeInt( n );
3618502788666131106986593281521497120414687020801267626233049500247285301313
gap> time;                     
1
gap> NextPrimeInt( n );        
3618502788666131106986593281521497120414687020801267626233049500247285301313
gap> time;             
213

4.1-5 PrimeNumbersIterator
‣ PrimeNumbersIterator( [chunksize] )( function )

This function has been transferred from package RCWA.

This function returns an iterator which runs over the prime numbers n ascending order; it takes an optional argument chunksize which specifies the length of the interval which is sieved in one go (the default is \(10^7\)), and which can be used to balance runtime vs. memory consumption. It is assumed that chunksize is larger than any gap between two consecutive primes within the range one intends to run the iterator over.


gap> iter := PrimeNumbersIterator();;
gap> for i in [1..100] do  p := NextIterator(iter);  od;
gap> p;
541
gap> sum := 0;;
gap> ## "prime number race" 1 vs. 3 mod 4
gap> for p in PrimeNumbersIterator() do 
>       if p <> 2 then sum := sum + E(4)^(p-1); fi;
>       if sum > 0 then break; fi;
>    od;
gap> p;
26861
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utils-0.86/doc/groups.xml000644 000765 000024 00000041071 14621444241 020526 0ustar00christopherwensleystaff000000 000000 Groups and homomorphisms
Functions for groups This method has been transferred from package &ResClasses;.

It provides a method for Comm when the argument is a list (enclosed in square brackets), and calls the function LeftNormedComm.

Comm( [ (1,2), (2,3) ] ); (1,2,3) gap> Comm( [(1,2),(2,3),(3,4),(4,5),(5,6)] ); (1,5,6) gap> Comm(Comm(Comm(Comm((1,2),(2,3)),(3,4)),(4,5)),(5,6)); ## the same (1,5,6) ]]> This function has been transferred from package &ResClasses;.

It tests whether two elements in a group commute.

D12 := DihedralGroup( 12 ); gap> SetName( D12, "D12" ); gap> a := D12.1;; b := D12.2;; gap> IsCommuting( a, b ); false ]]> This function has been transferred from package &RCWA;.

The operation ListOfPowers(g,exp) returns the list [g,g^2,...,g^{exp}] of powers of the element g.

ListOfPowers( 2, 20 ); [ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576 ] gap> ListOfPowers( (1,2,3)(4,5), 12 ); [ (1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), (), (1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), () ] gap> ListOfPowers( D12.2, 6 ); [ f2, f3, f2*f3, f3^2, f2*f3^2, of ... ] ]]> This function has been transferred from package &RCWA;.

This operation returns a list containing the generators of G followed by the inverses of these generators.

GeneratorsAndInverses( D12 ); [ f1, f2, f3, f1, f2*f3^2, f3^2 ] gap> GeneratorsAndInverses( SymmetricGroup(5) ); [ (1,2,3,4,5), (1,2), (1,5,4,3,2), (1,2) ] ]]> These three functions have been transferred from package &ResClasses;.

Fitting series The upper and lower Fitting series and the Fitting length of a solvable group are described here: https://en.wikipedia.org/wiki/Fitting_length.

UpperFittingSeries( D12 ); LowerFittingSeries( D12 ); [ Group([ ]), Group([ f3, f2*f3 ]), Group([ f1, f3, f2*f3 ]) ] [ D12, Group([ f3 ]), Group([ ]) ] gap> FittingLength( D12 ); 2 gap> S4 := SymmetricGroup( 4 );; gap> UpperFittingSeries( S4 ); [ Group(()), Group([ (1,2)(3,4), (1,4)(2,3) ]), Group([ (1,2)(3,4), (1,4) (2,3), (2,4,3) ]), Group([ (3,4), (2,3,4), (1,2)(3,4) ]) ] gap> List( last, StructureDescription ); [ "1", "C2 x C2", "A4", "S4" ] gap> LowerFittingSeries( S4 ); [ Sym( [ 1 .. 4 ] ), Alt( [ 1 .. 4 ] ), Group([ (1,4)(2,3), (1,3) (2,4) ]), Group(()) ] gap> List( last, StructureDescription ); [ "S4", "A4", "C2 x C2", "1" ] gap> FittingLength( S4); 3 ]]>

Left Cosets for Groups Since &GAP; uses right actions by default, the library contains the operation RightCoset(U,g) for constructing the right coset Ug of a subgroup U \leq G and an element g \in G. It has been noted in the reference manual that, by inverting all the elements in Ug, the left coset g^{-1}U is obtained.

Just for the sake of completeness, from August 2022 this package provides the operation LeftCoset(g,U) for constructing the left coset gU. Users are strongly recommended to continue to use RightCoset for all serious calculations, since left cosets have a much simpler implementation and do not behave exactly like right cosets.

The methods for left cosets which are provided generally work by converting gU to Ug^{-1}; applying the equivalent method for right cosets; and, if necessary, converting back again to left cosets.

G acts on gU by OnLeftInverse: (gU)^{g_0} = g_0^{-1}*(gU) = (g_0^{-1}g)U.

lc1 := LeftCoset( (1,2,3), Group( [ (1,2), (3,4) ] ) ); LeftCoset((1,2,3),Group([ (1,2), (3,4) ])) gap> Representative( lc1 ); (1,2,3) gap> ActingDomain( lc1 ); Group([ (1,2), (3,4) ]) gap> AsSet( lc1 ); [ (2,3), (2,4,3), (1,2,3), (1,2,4,3) ] gap> (1,2,3) in lc1; true gap> lc2 := (2,4,3) * lc1; LeftCoset((1,2,4),Group([ (1,2), (3,4) ])) gap> lc3 := lc1^(2,3,4);; gap> lc2 = lc3; true ]]> Inverse The inverse of the left coset gU is the right coset Ug^{-1}, and conversely. This is an abuse of the attribute Inverse, since the standard requirement, that x*x^{-1} is an identity, does not hold.

rc1 := Inverse( lc1 ); RightCoset(Group([ (1,2), (3,4) ]),(1,3,2)) gap> rc4 := RightCoset( Group( (1,2), (2,3) ), (3,4) ); RightCoset(Group([ (1,2), (2,3) ]),(3,4)) gap> lc4 := Inverse( rc4 ); LeftCoset((3,4),Group([ (1,2), (2,3) ])) gap> Intersection( lc2, lc4 ); [ (2,3,4), (1,2,3,4) ] ]]>

Functions for group homomorphisms This function has been transferred from package &RCWA;.

It constructs a group homomorphism which maps the generators of G to those of H. Its intended use is when G is a free group, and a warning is printed when this is not the case. Note that anything may happen if the resulting map is not a homomorphism!

G := Group( (1,2,3), (3,4,5), (5,6,7), (7,8,9) );; gap> phi := EpimorphismByGenerators( FreeGroup("a","b","c","d"), G ); [ a, b, c, d ] -> [ (1,2,3), (3,4,5), (5,6,7), (7,8,9) ] gap> PreImagesRepresentativeNC( phi, (1,2,3,4,5,6,7,8,9) ); d*c*b*a gap> a := G.1;; b := G.2;; c := G.3;; d := G.4;; gap> d*c*b*a; (1,2,3,4,5,6,7,8,9) gap> ## note that it is easy to produce nonsense: gap> epi := EpimorphismByGenerators( Group((1,2,3)), Group((8,9)) ); Warning: calling GroupHomomorphismByImagesNC without checks [ (1,2,3) ] -> [ (8,9) ] gap> IsGroupHomomorphism( epi ); true gap> Image( epi, (1,2,3) ); () gap> Image( epi, (1,3,2) ); (8,9) ]]> If \phi_1 : G_1 \to H and \phi_2 : G_2 \to H are two group homomorphisms with the same range, then their pullback is the subgroup of G_1 \times G_2 consisting of those elements (g_1,g_2) such that \phi_1 g_1 = \phi_2 g_2.

The attribute PullbackInfo of a pullback group P is similar to DirectProductInfo for a direct product of groups. Its value is a record with the following components:

directProduct the direct product G_1 \times G_2, and projections a list with the two projections onto G_1 and G_2. There are no embeddings in this record, but it is possible to use the embeddings into the direct product, see .

s4 := Group( (1,2),(2,3),(3,4) );; gap> s3 := Group( (5,6),(6,7) );; gap> c3 := Subgroup( s3, [ (5,6,7) ] );; gap> f := GroupHomomorphismByImages( s4, s3, > [(1,2),(2,3),(3,4)], [(5,6),(6,7),(5,6)] );; gap> i := GroupHomomorphismByImages( c3, s3, [(5,6,7)], [(5,6,7)] );; gap> Pfi := Pullback( f, i ); Group([ (2,3,4)(5,7,6), (1,2)(3,4) ]) gap> StructureDescription( Pfi ); "A4" gap> info := PullbackInfo( Pfi ); rec( directProduct := Group([ (1,2), (2,3), (3,4), (5,6,7) ]), projections := [ [ (2,3,4)(5,7,6), (1,2)(3,4) ] -> [ (2,3,4), (1,2)(3,4) ], [ (2,3,4)(5,7,6), (1,2)(3,4) ] -> [ (5,7,6), () ] ] ) gap> g := (1,2,3)(5,6,7);; gap> ImageElm( info!.projections[1], g ); (1,2,3) gap> ImageElm( info!.projections[2], g ); (5,6,7) gap> dp := info!.directProduct;; gap> a := ImageElm( Embedding( dp, 1 ), (1,4,3) );; gap> b := ImageElm( Embedding( dp, 2 ), (5,7,6) );; gap> a*b in Pfi; true ]]> This function was added by Thomas Breuer, following discussions with Hongyi Zhao (see https://github.com/gap-packages/hap/issues/73).

Let G1 and G2 be two groups, Z1 be a central subgroup of G1, and Phi be an isomorphism from Z1 to a central subgroup of G2. The central product defined by these arguments is the factor group of the direct product of G1 and G2 by the central subgroup \{ (z, (Phi(z))^{-1}) : z \in Z1 \}.

The attribute of a group G that has been created by is similar to for pullback groups. Its value is a record with the following components.

projection the epimorphism from the direct product of G1 and G2 to G, and phi the map Phi. Note that one can access the direct product as the value of the projection map, and one can access G1 and G2 as the two embeddings of this direct product, see . g1 := DihedralGroup( 8 ); gap> c1 := Centre( g1 ); Group([ f3 ]) gap> cp1 := CentralProduct( g1, g1, c1, IdentityMapping( c1 ) ); Group([ f1, f2, f5, f3, f4, f5 ]) gap> IdGroup( cp1 ) = IdGroup( ExtraspecialGroup( 2^5, "+" ) ); true gap> g2 := QuaternionGroup( 8 ); gap> c2 := Centre( g2 ); Group([ y2 ]) gap> cp2 := CentralProduct( g2, g2, c2, IdentityMapping( c2 ) ); Group([ f1, f2, f5, f3, f4, f5 ]) gap> IdGroup( cp2 ) = IdGroup( ExtraspecialGroup( 2^5, "+" ) ); true gap> info2 := CentralProductInfo( cp2 ); rec( phi := IdentityMapping( Group([ y2 ]) ), projection := [ f1, f2, f3, f4, f5, f6 ] -> [ f1, f2, f5, f3, f4, f5 ] ) gap> Source( Embedding( Source( info2.projection ), 1 ) ) = g2; true ]]> An endomorphism f : G \to G is idempotent if f^2=f. It has an image R \leqslant G; is the identity map when restricted to R; and has a kernel N which has trivial intersection with R and has size |G|/|R|.

The operation IdempotentEndomorphismsWithImage(genG,R) returns a list of the images of the generating set genG of a group G under the idempotent endomorphisms with image R.

The attribute IdempotentEndomorphismsData(G) returns a record data with fields data.gens, a fixed generating set for G, and data.images a list of the non-empty outputs of IdempotentEndomorphismsWithImage(genG,R) obtained by iterating over all subgroups of G.

The operation IdempotentEndomorphisms(G) returns the list of these mappings obtained using IdempotentEndomorphismsData(G). The first of these is the zero map, the second is the identity.

gens := [ (1,2,3,4), (1,2)(3,4) ];; gap> d8 := Group( gens );; gap> SetName( d8, "d8" ); gap> c2 := Subgroup( d8, [ (2,4) ] );; gap> IdempotentEndomorphismsWithImage( gens, c2 ); [ [ (), (2,4) ], [ (2,4), () ] ] gap> IdempotentEndomorphismsData( d8 ); rec( gens := [ (1,2,3,4), (1,2)(3,4) ], images := [ [ [ (), () ] ], [ [ (), (2,4) ], [ (2,4), () ] ], [ [ (), (1,3) ], [ (1,3), () ] ], [ [ (), (1,2)(3,4) ], [ (1,2)(3,4), (1,2)(3,4) ] ], [ [ (), (1,4)(2,3) ], [ (1,4)(2,3), (1,4)(2,3) ] ], [ [ (1,2,3,4), (1,2)(3,4) ] ] ] ) gap> List( last.images, L -> Length(L) ); [ 1, 2, 2, 2, 2, 1 ] gap> IdempotentEndomorphisms( d8 ); [ [ (1,2,3,4), (1,2)(3,4) ] -> [ (), () ], [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (2,4) ], [ (1,2,3,4), (1,2)(3,4) ] -> [ (2,4), () ], [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (1,3) ], [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,3), () ], [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (1,2)(3,4) ], [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,2)(3,4), (1,2)(3,4) ], [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (1,4)(2,3) ], [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,4)(2,3), (1,4)(2,3) ], [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,2,3,4), (1,2)(3,4) ] ] ]]> The quaternion group q8 is an example of a group with a tail: there is only one subgroup in the lattice which covers the identity subgroup. The only idempotent isomorphisms of such groups are the identity mapping and the zero mapping because the only pairs N,R are the whole group and the identity subgroup. q8 := QuaternionGroup( 8 );; gap> IdempotentEndomorphisms( q8 ); [ [ x, y ] -> [ of ..., of ... ], [ x, y ] -> [ x, y ] ] ]]> Given group homomorphisms f_1 : G_1 \to G_2 and f_2 : H_1 \to H_2, this operation return the product homomorphism f_1 \times f_2 : G_1 \times G_2 \to H_1 \times H_2.

c4 := Group( (1,2,3,4) );; gap> c2 := Group( (5,6) );; gap> f1 := GroupHomomorphismByImages( c4, c2, [(1,2,3,4)], [(5,6)] );; gap> c3 := Group( (1,2,3) );; gap> c6 := Group( (1,2,3,4,5,6) );; gap> f2 := GroupHomomorphismByImages( c3, c6, [(1,2,3)], [(1,3,5)(2,4,6)] );; gap> c4c3 := DirectProduct( c4, c3 ); Group([ (1,2,3,4), (5,6,7) ]) gap> c2c6 := DirectProduct( c2, c6 ); Group([ (1,2), (3,4,5,6,7,8) ]) gap> f := DirectProductOfFunctions( c4c3, c2c6, f1, f2 ); [ (1,2,3,4), (5,6,7) ] -> [ (1,2), (3,5,7)(4,6,8) ] gap> ImageElm( f, (1,4,3,2)(5,7,6) ); (1,2)(3,7,5)(4,8,6) ]]> Let A_1,A_2 be groups of automorphism of groups G_1,G_2 respectively. The output of this function is a group A_1 \times A_2 of automorphisms of G_1 \times G_2.

c9 := Group( (1,2,3,4,5,6,7,8,9) );; gap> ac9 := AutomorphismGroup( c9 );; gap> q8 := QuaternionGroup( IsPermGroup, 8 );; gap> aq8 := AutomorphismGroup( q8 );; gap> A := DirectProductOfAutomorphismGroups( ac9, aq8 ); gap> genA := GeneratorsOfGroup( A );; gap> G := Source( genA[1] ); Group([ (1,2,3,4,5,6,7,8,9), (10,14,12,16)(11,17,13,15), (10,11,12,13) (14,15,16,17) ]) gap> a := genA[1]*genA[5]; [ (1,2,3,4,5,6,7,8,9), (10,14,12,16)(11,17,13,15), (10,11,12,13)(14,15,16,17) ] -> [ (1,3,5,7,9,2,4,6,8), (10,16,12,14)(11,15,13,17), (10,11,12,13)(14,15,16,17) ] gap> ImageElm( a, (1,9,8,7,6,5,4,3,2)(10,14,12,16)(11,17,13,15) ); (1,8,6,4,2,9,7,5,3)(10,16,12,14)(11,15,13,17) ]]>

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} /* colors for ColorPrompt like examples */ span.GAPprompt { color: #000097; font-weight: normal; } span.GAPbrkprompt { color: #970000; font-weight: normal; } span.GAPinput { color: #970000; } /* Bib entries */ p.BibEntry { } span.BibKey { color: #005522; } span.BibKeyLink { } b.BibAuthor { } i.BibTitle { } i.BibBookTitle { } span.BibEditor { } span.BibJournal { } span.BibType { } span.BibPublisher { } span.BibSchool { } span.BibEdition { } span.BibVolume { } span.BibSeries { } span.BibNumber { } span.BibPages { } span.BibOrganization { } span.BibAddress { } span.BibYear { } span.BibPublisher { } span.BibNote { } span.BibHowpublished { } utils-0.86/doc/chap11_mj.html000644 000765 000024 00000017742 14621610515 021125 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Chapter 11: Obsolete functions
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11 Obsolete functions
 11.1 Operations from AutoDoc
 11.2 Functions for printing
 11.3 Other obsolete functions

  11.3-1 Applicable Methods
  11.3-2 ExponentOfPrime

11 Obsolete functions

11.1 Operations from AutoDoc

The file functions FindMatchingFiles and CreateDirIfMissing were copied from package AutoDoc where they are named AutoDoc_FindMatchingFiles and AutoDoc_CreateDirIfMissing.

The string function StringDotSuffix was also copied from package AutoDoc, where it is named AUTODOC_GetSuffix.

The function SetIfMissing was also transferred from package AutoDoc, where it is called AUTODOC_SetIfMissing. It writes into a record provided the position is not yet bound.

As from version 0.61, all these functions became obsolete in Utils, but continue to be defined in AutoDoc.

11.2 Functions for printing

The function PrintOneItemPerLine was used to prints lists vertically, rather than horizontally. Since a very similar result may be achieved using the GAP library functions Perform and Display, this function became obsolete in version 0.61.


gap> s3 := SymmetricGroup( 3 );; 
gap> L := KnownPropertiesOfObject( GeneratorsOfGroup( s3 ) );;
gap> Perform( L, Display );
IsFinite
IsSmallList
IsGeneratorsOfMagmaWithInverses
IsGeneratorsOfSemigroup
IsSubsetLocallyFiniteGroup
gap> Perform( s3, Display ); 
()
(2,3)
(1,3)
(1,3,2)
(1,2,3)
(1,2)

11.3 Other obsolete functions

11.3-1 Applicable Methods

The function PrintApplicableMethod, which was included in versions from 0.41 to 0.58, has been removed since it was considered superfluous. The example shows how to print out a function.


gap> ApplicableMethod( IsCyclic, [ Group((1,2,3),(4,5)) ], 1, 1 );
#I  Searching Method for IsCyclic with 1 arguments:
#I  Total: 7 entries
#I  Method 4: ``IsCyclic'' at /Applications/gap/gap4r9/lib/grp.gi:30 , value: 
36
function( G ) ... end
gap> Print( last );
function ( G )
    if Length( GeneratorsOfGroup( G ) ) = 1 then
        return true;
    else
        TryNextMethod();
    fi;
    return;
end
gap> ApplicableMethod( IsCyclic, [ Group((1,2,3),(4,5)) ], 0, 3 );
function( <1 unnamed arguments> ) ... end
gap> Print( last );                                               
function ( <<arg-1>> )
    <<compiled GAP code from GAPROOT/lib/oper1.g:578>>
end

11.3-2 ExponentOfPrime

The function ExponentOfPrime was originally transferred from package RCWA. The command ExponentOfPrime(n,p) returned the exponent of the prime p in the prime factorization of n.

Since the GAP function PValuation produces the same results, and does so more quickly, this function has been made obsolete.

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utils-0.86/doc/chap4.txt000644 000765 000024 00000016704 14621610512 020226 0ustar00christopherwensleystaff000000 000000 4 Number-theoretic functions 4.1 Functions for integers 4.1-1 AllSmoothIntegers AllSmoothIntegers( maxp, maxn )  function AllSmoothIntegers( L, maxp )  function This function has been transferred from package RCWA. The function AllSmoothIntegers(maxp,maxn) returns the list of all positive integers less than or equal to maxn whose prime factors are all in the list L = {p ~|~ p leqslant maxp, p~mboxprime }. In the alternative form, when L is a list of primes, the function returns the list of all positive integers whose prime factors lie in L.  Example   gap> AllSmoothIntegers( 3, 1000 ); [ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96,   108, 128, 144, 162, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576,   648, 729, 768, 864, 972 ] gap> AllSmoothIntegers( [5,11,17], 1000 ); [ 1, 5, 11, 17, 25, 55, 85, 121, 125, 187, 275, 289, 425, 605, 625, 935 ] gap> Length( last ); 16 gap> List( [3..20], n -> Length( AllSmoothIntegers( [5,11,17], 10^n ) ) ); [ 16, 29, 50, 78, 114, 155, 212, 282, 359, 452, 565, 691, 831, 992, 1173,   1374, 1595, 1843 ]   4.1-2 AllProducts AllProducts( L, k )  function This function has been transferred from package RCWA. The command AllProducts(L,k) returns the list of all products of k entries of the list L. Note that every ordering of the entries is used so that, in the commuting case, there are bound to be repetitions.  Example   gap> AllProducts([1..4],3);  [ 1, 2, 3, 4, 2, 4, 6, 8, 3, 6, 9, 12, 4, 8, 12, 16, 2, 4, 6, 8, 4, 8, 12,   16, 6, 12, 18, 24, 8, 16, 24, 32, 3, 6, 9, 12, 6, 12, 18, 24, 9, 18, 27,   36, 12, 24, 36, 48, 4, 8, 12, 16, 8, 16, 24, 32, 12, 24, 36, 48, 16, 32,   48, 64 ] gap> Set(last);  [ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 64 ] gap> AllProducts( [(1,2,3),(2,3,4)], 2 ); [ (2,4,3), (1,2)(3,4), (1,3)(2,4), (1,3,2) ]   4.1-3 RestrictedPartitionsWithoutRepetitions RestrictedPartitionsWithoutRepetitions( n, S )  function This function has been transferred from package RCWA. For a positive integer n and a set of positive integers S, this function returns the list of partitions of n into distinct elements of S. Unlike RestrictedPartitions, no repetitions are allowed.  Example   gap> RestrictedPartitions( 20, [4..10] ); [ [ 4, 4, 4, 4, 4 ], [ 5, 5, 5, 5 ], [ 6, 5, 5, 4 ], [ 6, 6, 4, 4 ],   [ 7, 5, 4, 4 ], [ 7, 7, 6 ], [ 8, 4, 4, 4 ], [ 8, 6, 6 ], [ 8, 7, 5 ],   [ 8, 8, 4 ], [ 9, 6, 5 ], [ 9, 7, 4 ], [ 10, 5, 5 ], [ 10, 6, 4 ],   [ 10, 10 ] ] gap> RestrictedPartitionsWithoutRepetitions( 20, [4..10] ); [ [ 10, 6, 4 ], [ 9, 7, 4 ], [ 9, 6, 5 ], [ 8, 7, 5 ] ] gap> RestrictedPartitionsWithoutRepetitions( 10^2, List([1..10], n->n^2 ) ); [ [ 100 ], [ 64, 36 ], [ 49, 25, 16, 9, 1 ] ]   4.1-4 NextProbablyPrimeInt NextProbablyPrimeInt( n )  function This function has been transferred from package RCWA. The function NextProbablyPrimeInt(n) does the same as NextPrimeInt(n) except that for reasons of performance it tests numbers only for IsProbablyPrimeInt(n) instead of IsPrimeInt(n). For large n, this function is much faster than NextPrimeInt(n)  Example   gap> n := 2^251; 3618502788666131106986593281521497120414687020801267626233049500247285301248 gap> NextProbablyPrimeInt( n ); 3618502788666131106986593281521497120414687020801267626233049500247285301313 gap> time;  1 gap> NextPrimeInt( n );  3618502788666131106986593281521497120414687020801267626233049500247285301313 gap> time;  213   4.1-5 PrimeNumbersIterator PrimeNumbersIterator( [chunksize] )  function This function has been transferred from package RCWA. This function returns an iterator which runs over the prime numbers n ascending order; it takes an optional argument chunksize which specifies the length of the interval which is sieved in one go (the default is 10^7), and which can be used to balance runtime vs. memory consumption. It is assumed that chunksize is larger than any gap between two consecutive primes within the range one intends to run the iterator over.  Example   gap> iter := PrimeNumbersIterator();; gap> for i in [1..100] do p := NextIterator(iter); od; gap> p; 541 gap> sum := 0;; gap> ## "prime number race" 1 vs. 3 mod 4 gap> for p in PrimeNumbersIterator() do  >  if p <> 2 then sum := sum + E(4)^(p-1); fi; >  if sum > 0 then break; fi; >  od; gap> p; 26861   utils-0.86/doc/chap5.txt000644 000765 000024 00000057727 14621610512 020241 0ustar00christopherwensleystaff000000 000000 5 Groups and homomorphisms 5.1 Functions for groups 5.1-1 Comm Comm( L )  operation This method has been transferred from package ResClasses. It provides a method for Comm when the argument is a list (enclosed in square brackets), and calls the function LeftNormedComm.  Example   gap> Comm( [ (1,2), (2,3) ] ); (1,2,3) gap> Comm( [(1,2),(2,3),(3,4),(4,5),(5,6)] ); (1,5,6) gap> Comm(Comm(Comm(Comm((1,2),(2,3)),(3,4)),(4,5)),(5,6)); ## the same (1,5,6)   5.1-2 IsCommuting IsCommuting( a, b )  operation This function has been transferred from package ResClasses. It tests whether two elements in a group commute.  Example   gap> D12 := DihedralGroup( 12 );  gap> SetName( D12, "D12" );  gap> a := D12.1;; b := D12.2;;  gap> IsCommuting( a, b ); false   5.1-3 ListOfPowers ListOfPowers( g, exp )  operation This function has been transferred from package RCWA. The operation ListOfPowers(g,exp) returns the list [g,g^2,...,g^exp] of powers of the element g.  Example   gap> ListOfPowers( 2, 20 ); [ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384,  32768, 65536, 131072, 262144, 524288, 1048576 ] gap> ListOfPowers( (1,2,3)(4,5), 12 ); [ (1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), (),  (1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), () ] gap> ListOfPowers( D12.2, 6 ); [ f2, f3, f2*f3, f3^2, f2*f3^2, of ... ]   5.1-4 GeneratorsAndInverses GeneratorsAndInverses( G )  operation This function has been transferred from package RCWA. This operation returns a list containing the generators of G followed by the inverses of these generators.  Example   gap> GeneratorsAndInverses( D12 ); [ f1, f2, f3, f1, f2*f3^2, f3^2 ] gap> GeneratorsAndInverses( SymmetricGroup(5) );  [ (1,2,3,4,5), (1,2), (1,5,4,3,2), (1,2) ]   5.1-5 UpperFittingSeries UpperFittingSeries( G )  attribute LowerFittingSeries( G )  attribute FittingLength( G )  attribute These three functions have been transferred from package ResClasses. The upper and lower Fitting series and the Fitting length of a solvable group are described here: https://en.wikipedia.org/wiki/Fitting_length.  Example   gap> UpperFittingSeries( D12 ); LowerFittingSeries( D12 ); [ Group([ ]), Group([ f3, f2*f3 ]), Group([ f1, f3, f2*f3 ]) ] [ D12, Group([ f3 ]), Group([ ]) ] gap> FittingLength( D12 ); 2 gap> S4 := SymmetricGroup( 4 );; gap> UpperFittingSeries( S4 ); [ Group(()), Group([ (1,2)(3,4), (1,4)(2,3) ]), Group([ (1,2)(3,4), (1,4)  (2,3), (2,4,3) ]), Group([ (3,4), (2,3,4), (1,2)(3,4) ]) ] gap> List( last, StructureDescription ); [ "1", "C2 x C2", "A4", "S4" ] gap> LowerFittingSeries( S4 ); [ Sym( [ 1 .. 4 ] ), Alt( [ 1 .. 4 ] ), Group([ (1,4)(2,3), (1,3)  (2,4) ]), Group(()) ] gap> List( last, StructureDescription ); [ "S4", "A4", "C2 x C2", "1" ] gap> FittingLength( S4); 3   5.2 Left Cosets for Groups 5.2-1 LeftCoset LeftCoset( g, U )  operation Since GAP uses right actions by default, the library contains the operation RightCoset(U,g) for constructing the right coset Ug of a subgroup U ≤ G and an element g ∈ G. It has been noted in the reference manual that, by inverting all the elements in Ug, the left coset g^-1U is obtained. Just for the sake of completeness, from August 2022 this package provides the operation LeftCoset(g,U) for constructing the left coset gU. Users are strongly recommended to continue to use RightCoset for all serious calculations, since left cosets have a much simpler implementation and do not behave exactly like right cosets. The methods for left cosets which are provided generally work by converting gU to Ug^-1; applying the equivalent method for right cosets; and, if necessary, converting back again to left cosets. G acts on gU by OnLeftInverse: (gU)^g_0 = g_0^-1*(gU) = (g_0^-1g)U.  Example   gap> lc1 := LeftCoset( (1,2,3), Group( [ (1,2), (3,4) ] ) );  LeftCoset((1,2,3),Group([ (1,2), (3,4) ])) gap> Representative( lc1 ); (1,2,3) gap> ActingDomain( lc1 ); Group([ (1,2), (3,4) ]) gap> AsSet( lc1 ); [ (2,3), (2,4,3), (1,2,3), (1,2,4,3) ] gap> (1,2,3) in lc1; true gap> lc2 := (2,4,3) * lc1;  LeftCoset((1,2,4),Group([ (1,2), (3,4) ])) gap> lc3 := lc1^(2,3,4);; gap> lc2 = lc3;  true   5.2-2 Inverse The inverse of the left coset gU is the right coset Ug^-1, and conversely. This is an abuse of the attribute Inverse, since the standard requirement, that x*x^-1 is an identity, does not hold.  Example   gap> rc1 := Inverse( lc1 );  RightCoset(Group([ (1,2), (3,4) ]),(1,3,2)) gap> rc4 := RightCoset( Group( (1,2), (2,3) ), (3,4) );  RightCoset(Group([ (1,2), (2,3) ]),(3,4)) gap> lc4 := Inverse( rc4 ); LeftCoset((3,4),Group([ (1,2), (2,3) ])) gap> Intersection( lc2, lc4 ); [ (2,3,4), (1,2,3,4) ]   5.3 Functions for group homomorphisms 5.3-1 EpimorphismByGenerators EpimorphismByGenerators( G, H )  operation This function has been transferred from package RCWA. It constructs a group homomorphism which maps the generators of G to those of H. Its intended use is when G is a free group, and a warning is printed when this is not the case. Note that anything may happen if the resulting map is not a homomorphism!  Example   gap> G := Group( (1,2,3), (3,4,5), (5,6,7), (7,8,9) );; gap> phi := EpimorphismByGenerators( FreeGroup("a","b","c","d"), G ); [ a, b, c, d ] -> [ (1,2,3), (3,4,5), (5,6,7), (7,8,9) ] gap> PreImagesRepresentativeNC( phi, (1,2,3,4,5,6,7,8,9) ); d*c*b*a gap> a := G.1;; b := G.2;; c := G.3;; d := G.4;; gap> d*c*b*a; (1,2,3,4,5,6,7,8,9) gap> ## note that it is easy to produce nonsense:  gap> epi := EpimorphismByGenerators( Group((1,2,3)), Group((8,9)) ); Warning: calling GroupHomomorphismByImagesNC without checks [ (1,2,3) ] -> [ (8,9) ] gap> IsGroupHomomorphism( epi ); true gap> Image( epi, (1,2,3) );  () gap> Image( epi, (1,3,2) ); (8,9)   5.3-2 Pullback Pullback( hom1, hom2 )  operation PullbackInfo( G )  attribute If ϕ_1 : G_1 -> H and ϕ_2 : G_2 -> H are two group homomorphisms with the same range, then their pullback is the subgroup of G_1 × G_2 consisting of those elements (g_1,g_2) such that ϕ_1 g_1 = ϕ_2 g_2. The attribute PullbackInfo of a pullback group P is similar to DirectProductInfo for a direct product of groups. Its value is a record with the following components: directProduct the direct product G_1 × G_2, and projections a list with the two projections onto G_1 and G_2. There are no embeddings in this record, but it is possible to use the embeddings into the direct product, see Embedding (Reference: Embedding).  Example   gap> s4 := Group( (1,2),(2,3),(3,4) );; gap> s3 := Group( (5,6),(6,7) );; gap> c3 := Subgroup( s3, [ (5,6,7) ] );; gap> f := GroupHomomorphismByImages( s4, s3,  >  [(1,2),(2,3),(3,4)], [(5,6),(6,7),(5,6)] );;  gap> i := GroupHomomorphismByImages( c3, s3, [(5,6,7)], [(5,6,7)] );;  gap> Pfi := Pullback( f, i ); Group([ (2,3,4)(5,7,6), (1,2)(3,4) ]) gap> StructureDescription( Pfi ); "A4" gap> info := PullbackInfo( Pfi ); rec( directProduct := Group([ (1,2), (2,3), (3,4), (5,6,7) ]),   projections := [ [ (2,3,4)(5,7,6), (1,2)(3,4) ] -> [ (2,3,4), (1,2)(3,4) ],   [ (2,3,4)(5,7,6), (1,2)(3,4) ] -> [ (5,7,6), () ] ] ) gap> g := (1,2,3)(5,6,7);;  gap> ImageElm( info!.projections[1], g ); (1,2,3) gap> ImageElm( info!.projections[2], g ); (5,6,7)  gap> dp := info!.directProduct;;  gap> a := ImageElm( Embedding( dp, 1 ), (1,4,3) );;  gap> b := ImageElm( Embedding( dp, 2 ), (5,7,6) );;  gap> a*b in Pfi; true   5.3-3 CentralProduct CentralProduct( G1, G2, Z1, Phi )  operation CentralProductInfo( G )  attribute This function was added by Thomas Breuer, following discussions with Hongyi Zhao (see https://github.com/gap-packages/hap/issues/73). Let G1 and G2 be two groups, Z1 be a central subgroup of G1, and Phi be an isomorphism from Z1 to a central subgroup of G2. The central product defined by these arguments is the factor group of the direct product of G1 and G2 by the central subgroup { (z, (Phi(z))^-1) : z ∈ Z1 }. The attribute CentralProductInfo of a group G that has been created by CentralProduct is similar to PullbackInfo (5.3-2) for pullback groups. Its value is a record with the following components. projection the epimorphism from the direct product of G1 and G2 to G, and phi the map Phi. Note that one can access the direct product as the Source (Reference: Source) value of the projection map, and one can access G1 and G2 as the two embeddings of this direct product, see Embedding (Reference: Embedding).  Example  gap> g1 := DihedralGroup( 8 );  gap> c1 := Centre( g1 ); Group([ f3 ]) gap> cp1 := CentralProduct( g1, g1, c1, IdentityMapping( c1 ) ); Group([ f1, f2, f5, f3, f4, f5 ]) gap> IdGroup( cp1 ) = IdGroup( ExtraspecialGroup( 2^5, "+" ) ); true gap> g2 := QuaternionGroup( 8 );  gap> c2 := Centre( g2 ); Group([ y2 ]) gap> cp2 := CentralProduct( g2, g2, c2, IdentityMapping( c2 ) ); Group([ f1, f2, f5, f3, f4, f5 ]) gap> IdGroup( cp2 ) = IdGroup( ExtraspecialGroup( 2^5, "+" ) ); true gap> info2 := CentralProductInfo( cp2 ); rec( phi := IdentityMapping( Group([ y2 ]) ),   projection := [ f1, f2, f3, f4, f5, f6 ] -> [ f1, f2, f5, f3, f4, f5 ] ) gap> Source( Embedding( Source( info2.projection ), 1 ) ) = g2; true  5.3-4 IdempotentEndomorphisms IdempotentEndomorphisms( G )  operation IdempotentEndomorphismsData( G )  attribute IdempotentEndomorphismsWithImage( genG, R )  operation An endomorphism f : G -> G is idempotent if f^2=f. It has an image R leqslant G; is the identity map when restricted to R; and has a kernel N which has trivial intersection with R and has size |G|/|R|. The operation IdempotentEndomorphismsWithImage(genG,R) returns a list of the images of the generating set genG of a group G under the idempotent endomorphisms with image R. The attribute IdempotentEndomorphismsData(G) returns a record data with fields data.gens, a fixed generating set for G, and data.images a list of the non-empty outputs of IdempotentEndomorphismsWithImage(genG,R) obtained by iterating over all subgroups of G. The operation IdempotentEndomorphisms(G) returns the list of these mappings obtained using IdempotentEndomorphismsData(G). The first of these is the zero map, the second is the identity.  Example   gap> gens := [ (1,2,3,4), (1,2)(3,4) ];;  gap> d8 := Group( gens );; gap> SetName( d8, "d8" ); gap> c2 := Subgroup( d8, [ (2,4) ] );; gap> IdempotentEndomorphismsWithImage( gens, c2 ); [ [ (), (2,4) ], [ (2,4), () ] ] gap> IdempotentEndomorphismsData( d8 ); rec( gens := [ (1,2,3,4), (1,2)(3,4) ],   images := [ [ [ (), () ] ], [ [ (), (2,4) ], [ (2,4), () ] ],   [ [ (), (1,3) ], [ (1,3), () ] ],   [ [ (), (1,2)(3,4) ], [ (1,2)(3,4), (1,2)(3,4) ] ],   [ [ (), (1,4)(2,3) ], [ (1,4)(2,3), (1,4)(2,3) ] ],   [ [ (1,2,3,4), (1,2)(3,4) ] ] ] ) gap> List( last.images, L -> Length(L) ); [ 1, 2, 2, 2, 2, 1 ] gap> IdempotentEndomorphisms( d8 );  [ [ (1,2,3,4), (1,2)(3,4) ] -> [ (), () ],   [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (2,4) ],   [ (1,2,3,4), (1,2)(3,4) ] -> [ (2,4), () ],   [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (1,3) ],   [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,3), () ],   [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (1,2)(3,4) ],   [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,2)(3,4), (1,2)(3,4) ],   [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (1,4)(2,3) ],   [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,4)(2,3), (1,4)(2,3) ],   [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,2,3,4), (1,2)(3,4) ] ]   The quaternion group q8 is an example of a group with a tail: there is only one subgroup in the lattice which covers the identity subgroup. The only idempotent isomorphisms of such groups are the identity mapping and the zero mapping because the only pairs N,R are the whole group and the identity subgroup.  Example   gap> q8 := QuaternionGroup( 8 );; gap> IdempotentEndomorphisms( q8 ); [ [ x, y ] -> [ of ..., of ... ], [ x, y ] -> [ x, y ] ]   5.3-5 DirectProductOfFunctions DirectProductOfFunctions( G, H, f1, f2 )  operation Given group homomorphisms f_1 : G_1 -> G_2 and f_2 : H_1 -> H_2, this operation return the product homomorphism f_1 × f_2 : G_1 × G_2 -> H_1 × H_2.  Example   gap> c4 := Group( (1,2,3,4) );;  gap> c2 := Group( (5,6) );;  gap> f1 := GroupHomomorphismByImages( c4, c2, [(1,2,3,4)], [(5,6)] );; gap> c3 := Group( (1,2,3) );;  gap> c6 := Group( (1,2,3,4,5,6) );;  gap> f2 := GroupHomomorphismByImages( c3, c6, [(1,2,3)], [(1,3,5)(2,4,6)] );;  gap> c4c3 := DirectProduct( c4, c3 );  Group([ (1,2,3,4), (5,6,7) ]) gap> c2c6 := DirectProduct( c2, c6 );  Group([ (1,2), (3,4,5,6,7,8) ]) gap> f := DirectProductOfFunctions( c4c3, c2c6, f1, f2 );  [ (1,2,3,4), (5,6,7) ] -> [ (1,2), (3,5,7)(4,6,8) ] gap> ImageElm( f, (1,4,3,2)(5,7,6) );  (1,2)(3,7,5)(4,8,6)   5.3-6 DirectProductOfAutomorphismGroups DirectProductOfAutomorphismGroups( A1, A2 )  operation Let A_1,A_2 be groups of automorphism of groups G_1,G_2 respectively. The output of this function is a group A_1 × A_2 of automorphisms of G_1 × G_2.  Example   gap> c9 := Group( (1,2,3,4,5,6,7,8,9) );;  gap> ac9 := AutomorphismGroup( c9 );;  gap> q8 := QuaternionGroup( IsPermGroup, 8 );; gap> aq8 := AutomorphismGroup( q8 );; gap> A := DirectProductOfAutomorphismGroups( ac9, aq8 );  gap> genA := GeneratorsOfGroup( A );; gap> G := Source( genA[1] ); Group([ (1,2,3,4,5,6,7,8,9), (10,14,12,16)(11,17,13,15), (10,11,12,13) (14,15,16,17) ]) gap> a := genA[1]*genA[5];  [ (1,2,3,4,5,6,7,8,9), (10,14,12,16)(11,17,13,15), (10,11,12,13)(14,15,16,17)   ] -> [ (1,3,5,7,9,2,4,6,8), (10,16,12,14)(11,15,13,17),   (10,11,12,13)(14,15,16,17) ] gap> ImageElm( a, (1,9,8,7,6,5,4,3,2)(10,14,12,16)(11,17,13,15) ); (1,8,6,4,2,9,7,5,3)(10,16,12,14)(11,15,13,17)   utils-0.86/doc/ragged.css000644 000765 000024 00000000231 14621610515 020420 0ustar00christopherwensleystaff000000 000000 /* times.css Frank Lübeck */ /* Change default CSS to use Times font. */ body { text-align: left; } utils-0.86/doc/chap1.html000644 000765 000024 00000020726 14621610515 020352 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Chapter 1: Introduction
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1 Introduction
 1.1 Information for package authors

1 Introduction

The Utils package provides a space for utility functions from a variety of GAP packages to be collected together into a single package. In this way it is hoped that they will become more visible to other package authors. This package was first distributed as part of the GAP 4.8.2 distribution.

The package is loaded with the command


gap> LoadPackage( "utils" );

Functions have been transferred from the following packages:

  • Conversion of a GAP group to a Magma output string, taken from various sources including other.gi in the main library.

Transfer is complete (for now) for functions from the following packages:

The package may be obtained either as a compressed .tar file or as a .zip file, utils-version_number.tar.gz, by ftp from one of the following sites:

The package also has a GitHub repository at: https://github.com/gap-packages/utils.

Once the package is loaded, the manual doc/manual.pdf can be found in the documentation folder. The html versions, with or without MathJax, may be rebuilt as follows:


gap> ReadPackage( "utils", "makedoc.g" );

It is possible to check that the package has been installed correctly by running the test files (which terminates the GAP session):


gap> ReadPackage( "utils", "tst/testall.g" );
Architecture: . . . . . 
testing: . . . . . 
. . . 
#I  No errors detected while testing

Note that functions listed in this manual that are currently in the process of being transferred are only read from the source package Home (say), and so can only be used if Home has already been loaded. There are no such functions in transition at present.

1.1 Information for package authors

A function (or collection of functions) is suitable for transfer from a package Home to Utils if the following conditions are satisfied.

  • The function is sufficiently non-specialised so that it might be of use to other authors.

  • The function does not depend on the remaining functions in Home

  • The function does not do what can already be done with a GAP library function.

  • Documentation of the function and test examples are available.

  • When there is more than one active author of Home, they should all be aware (and content) that the transfer is taking place.

Authors of packages may be reluctant to let go of their utility functions. The following principles may help to reassure them. (Suggestions for more items here are welcome.)

  • A function that has been transferred to Utils will not be changed without the approval of the original author.

  • The current package maintainer has every intention of continuing to maintain Utils. In the event that this proves impossible, the GAP development team will surely find someone to take over.

  • Function names will not be changed unless specifically requested by Home's author(s) or unless they have the form HOME_FunctionName.

  • In order to speed up the transfer process, only functions from one package will be in transition at any given time. Hopefully a week or two will suffice for most packages.

  • Any package author who transfers a function to Utils will become an author of Utils. (In truth, Utils does not have authors, just a large number of contributors.)

The process for transferring utility functions from Home to Utils is described in Chapter 12.

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5 Groups and homomorphisms
 5.1 Functions for groups

  5.1-1 Comm
  5.1-2 IsCommuting
  5.1-3 ListOfPowers
  5.1-4 GeneratorsAndInverses
  5.1-5 UpperFittingSeries
 5.2 Left Cosets for Groups

  5.2-1 LeftCoset
  5.2-2 Inverse
 5.3 Functions for group homomorphisms

  5.3-1 EpimorphismByGenerators
  5.3-2 Pullback
  5.3-3 CentralProduct
  5.3-4 IdempotentEndomorphisms
  5.3-5 DirectProductOfFunctions
  5.3-6 DirectProductOfAutomorphismGroups

5 Groups and homomorphisms

5.1 Functions for groups

5.1-1 Comm
‣ Comm( L )( operation )

This method has been transferred from package ResClasses.

It provides a method for Comm when the argument is a list (enclosed in square brackets), and calls the function LeftNormedComm.


gap> Comm( [ (1,2), (2,3) ] );
(1,2,3)
gap> Comm( [(1,2),(2,3),(3,4),(4,5),(5,6)] );
(1,5,6)
gap> Comm(Comm(Comm(Comm((1,2),(2,3)),(3,4)),(4,5)),(5,6));  ## the same
(1,5,6)

5.1-2 IsCommuting
‣ IsCommuting( a, b )( operation )

This function has been transferred from package ResClasses.

It tests whether two elements in a group commute.


gap> D12 := DihedralGroup( 12 );
<pc group of size 12 with 3 generators>
gap> SetName( D12, "D12" ); 
gap> a := D12.1;;  b := D12.2;;  
gap> IsCommuting( a, b );
false

5.1-3 ListOfPowers
‣ ListOfPowers( g, exp )( operation )

This function has been transferred from package RCWA.

The operation ListOfPowers(g,exp) returns the list \([g,g^2,...,g^{exp}]\) of powers of the element \(g\).


gap> ListOfPowers( 2, 20 );
[ 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384,
 32768, 65536, 131072, 262144, 524288, 1048576 ]
gap> ListOfPowers( (1,2,3)(4,5), 12 );
[ (1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), (),
 (1,2,3)(4,5), (1,3,2), (4,5), (1,2,3), (1,3,2)(4,5), () ]
gap> ListOfPowers( D12.2, 6 );
[ f2, f3, f2*f3, f3^2, f2*f3^2, <identity> of ... ]

5.1-4 GeneratorsAndInverses
‣ GeneratorsAndInverses( G )( operation )

This function has been transferred from package RCWA.

This operation returns a list containing the generators of \(G\) followed by the inverses of these generators.


gap> GeneratorsAndInverses( D12 );
[ f1, f2, f3, f1, f2*f3^2, f3^2 ]
gap> GeneratorsAndInverses( SymmetricGroup(5) );     
[ (1,2,3,4,5), (1,2), (1,5,4,3,2), (1,2) ]

5.1-5 UpperFittingSeries
‣ UpperFittingSeries( G )( attribute )
‣ LowerFittingSeries( G )( attribute )
‣ FittingLength( G )( attribute )

These three functions have been transferred from package ResClasses.

The upper and lower Fitting series and the Fitting length of a solvable group are described here: https://en.wikipedia.org/wiki/Fitting_length.


gap> UpperFittingSeries( D12 );  LowerFittingSeries( D12 );
[ Group([  ]), Group([ f3, f2*f3 ]), Group([ f1, f3, f2*f3 ]) ]
[ D12, Group([ f3 ]), Group([  ]) ]
gap> FittingLength( D12 );
2
gap> S4 := SymmetricGroup( 4 );;
gap> UpperFittingSeries( S4 );
[ Group(()), Group([ (1,2)(3,4), (1,4)(2,3) ]), Group([ (1,2)(3,4), (1,4)
  (2,3), (2,4,3) ]), Group([ (3,4), (2,3,4), (1,2)(3,4) ]) ]
gap> List( last, StructureDescription );
[ "1", "C2 x C2", "A4", "S4" ]
gap> LowerFittingSeries( S4 );
[ Sym( [ 1 .. 4 ] ), Alt( [ 1 .. 4 ] ), Group([ (1,4)(2,3), (1,3)
 (2,4) ]), Group(()) ]
gap> List( last, StructureDescription );
[ "S4", "A4", "C2 x C2", "1" ]
gap> FittingLength( S4);
3

5.2 Left Cosets for Groups

5.2-1 LeftCoset
‣ LeftCoset( g, U )( operation )

Since GAP uses right actions by default, the library contains the operation RightCoset(U,g) for constructing the right coset \(Ug\) of a subgroup \(U \leq G\) and an element \(g \in G\). It has been noted in the reference manual that, by inverting all the elements in \(Ug\), the left coset \(g^{-1}U\) is obtained.

Just for the sake of completeness, from August 2022 this package provides the operation LeftCoset(g,U) for constructing the left coset \(gU\). Users are strongly recommended to continue to use RightCoset for all serious calculations, since left cosets have a much simpler implementation and do not behave exactly like right cosets.

The methods for left cosets which are provided generally work by converting \(gU\) to \(Ug^{-1}\); applying the equivalent method for right cosets; and, if necessary, converting back again to left cosets.

\(G\) acts on \(gU\) by OnLeftInverse: \((gU)^{g_0} = g_0^{-1}*(gU) = (g_0^{-1}g)U\).


gap> lc1 := LeftCoset( (1,2,3), Group( [ (1,2), (3,4) ] ) ); 
LeftCoset((1,2,3),Group([ (1,2), (3,4) ]))
gap> Representative( lc1 );
(1,2,3)
gap> ActingDomain( lc1 );
Group([ (1,2), (3,4) ])
gap> AsSet( lc1 );
[ (2,3), (2,4,3), (1,2,3), (1,2,4,3) ]
gap> (1,2,3) in lc1;
true
gap> lc2 := (2,4,3) * lc1;   
LeftCoset((1,2,4),Group([ (1,2), (3,4) ]))
gap> lc3 := lc1^(2,3,4);;
gap> lc2 = lc3;            
true

5.2-2 Inverse

The inverse of the left coset \(gU\) is the right coset \(Ug^{-1}\), and conversely. This is an abuse of the attribute Inverse, since the standard requirement, that \(x*x^{-1}\) is an identity, does not hold.


gap> rc1 := Inverse( lc1 ); 
RightCoset(Group([ (1,2), (3,4) ]),(1,3,2))
gap> rc4 := RightCoset( Group( (1,2), (2,3) ), (3,4) ); 
RightCoset(Group([ (1,2), (2,3) ]),(3,4))
gap> lc4 := Inverse( rc4 );
LeftCoset((3,4),Group([ (1,2), (2,3) ]))
gap> Intersection( lc2, lc4 );
[ (2,3,4), (1,2,3,4) ]

5.3 Functions for group homomorphisms

5.3-1 EpimorphismByGenerators
‣ EpimorphismByGenerators( G, H )( operation )

This function has been transferred from package RCWA.

It constructs a group homomorphism which maps the generators of \(G\) to those of \(H\). Its intended use is when \(G\) is a free group, and a warning is printed when this is not the case. Note that anything may happen if the resulting map is not a homomorphism!


gap> G := Group( (1,2,3), (3,4,5), (5,6,7), (7,8,9) );;
gap> phi := EpimorphismByGenerators( FreeGroup("a","b","c","d"), G );
[ a, b, c, d ] -> [ (1,2,3), (3,4,5), (5,6,7), (7,8,9) ]
gap> PreImagesRepresentativeNC( phi, (1,2,3,4,5,6,7,8,9) );
d*c*b*a
gap> a := G.1;; b := G.2;; c := G.3;; d := G.4;;
gap> d*c*b*a;
(1,2,3,4,5,6,7,8,9)
gap> ## note that it is easy to produce nonsense: 
gap> epi := EpimorphismByGenerators( Group((1,2,3)), Group((8,9)) );
Warning: calling GroupHomomorphismByImagesNC without checks
[ (1,2,3) ] -> [ (8,9) ]
gap> IsGroupHomomorphism( epi );
true
gap> Image( epi, (1,2,3) );                                            
()
gap> Image( epi, (1,3,2) );
(8,9)

5.3-2 Pullback
‣ Pullback( hom1, hom2 )( operation )
‣ PullbackInfo( G )( attribute )

If \(\phi_1 : G_1 \to H\) and \(\phi_2 : G_2 \to H\) are two group homomorphisms with the same range, then their pullback is the subgroup of \(G_1 \times G_2\) consisting of those elements \((g_1,g_2)\) such that \(\phi_1 g_1 = \phi_2 g_2\).

The attribute PullbackInfo of a pullback group P is similar to DirectProductInfo for a direct product of groups. Its value is a record with the following components:

directProduct

the direct product \(G_1 \times G_2\), and

projections

a list with the two projections onto \(G_1\) and \(G_2\).

There are no embeddings in this record, but it is possible to use the embeddings into the direct product, see Embedding (Reference: Embedding).


gap> s4 := Group( (1,2),(2,3),(3,4) );;
gap> s3 := Group( (5,6),(6,7) );;
gap> c3 := Subgroup( s3, [ (5,6,7) ] );;
gap> f := GroupHomomorphismByImages( s4, s3, 
>             [(1,2),(2,3),(3,4)], [(5,6),(6,7),(5,6)] );; 
gap> i := GroupHomomorphismByImages( c3, s3, [(5,6,7)], [(5,6,7)] );; 
gap> Pfi := Pullback( f, i );
Group([ (2,3,4)(5,7,6), (1,2)(3,4) ])
gap> StructureDescription( Pfi );
"A4"
gap> info := PullbackInfo( Pfi );
rec( directProduct := Group([ (1,2), (2,3), (3,4), (5,6,7) ]), 
  projections := [ [ (2,3,4)(5,7,6), (1,2)(3,4) ] -> [ (2,3,4), (1,2)(3,4) ], 
      [ (2,3,4)(5,7,6), (1,2)(3,4) ] -> [ (5,7,6), () ] ] )
gap> g := (1,2,3)(5,6,7);; 
gap> ImageElm( info!.projections[1], g );
(1,2,3)
gap> ImageElm( info!.projections[2], g );
(5,6,7) 
gap> dp := info!.directProduct;; 
gap> a := ImageElm( Embedding( dp, 1 ), (1,4,3) );; 
gap> b := ImageElm( Embedding( dp, 2 ), (5,7,6) );; 
gap> a*b in Pfi;
true

5.3-3 CentralProduct
‣ CentralProduct( G1, G2, Z1, Phi )( operation )
‣ CentralProductInfo( G )( attribute )

This function was added by Thomas Breuer, following discussions with Hongyi Zhao (see https://github.com/gap-packages/hap/issues/73).

Let G1 and G2 be two groups, Z1 be a central subgroup of G1, and Phi be an isomorphism from Z1 to a central subgroup of G2. The central product defined by these arguments is the factor group of the direct product of G1 and G2 by the central subgroup \(\{ (z, (\textit{Phi}(z))^{-1}) : z \in \textit{Z1} \}\).

The attribute CentralProductInfo of a group \(G\) that has been created by CentralProduct is similar to PullbackInfo (5.3-2) for pullback groups. Its value is a record with the following components.

projection

the epimorphism from the direct product of G1 and G2 to \(G\), and

phi

the map Phi.

Note that one can access the direct product as the Source (Reference: Source) value of the projection map, and one can access G1 and G2 as the two embeddings of this direct product, see Embedding (Reference: Embedding).

gap> g1 := DihedralGroup( 8 );
<pc group of size 8 with 3 generators>
gap> c1 := Centre( g1 );
Group([ f3 ])
gap> cp1 := CentralProduct( g1, g1, c1, IdentityMapping( c1 ) );
Group([ f1, f2, f5, f3, f4, f5 ])
gap> IdGroup( cp1 ) = IdGroup( ExtraspecialGroup( 2^5, "+" ) );
true
gap> g2 := QuaternionGroup( 8 );
<pc group of size 8 with 3 generators>
gap> c2 := Centre( g2 );
Group([ y2 ])
gap> cp2 := CentralProduct( g2, g2, c2, IdentityMapping( c2 ) );
Group([ f1, f2, f5, f3, f4, f5 ])
gap> IdGroup( cp2 ) = IdGroup( ExtraspecialGroup( 2^5, "+" ) );
true
gap> info2 := CentralProductInfo( cp2 );
rec( phi := IdentityMapping( Group([ y2 ]) ), 
  projection := [ f1, f2, f3, f4, f5, f6 ] -> [ f1, f2, f5, f3, f4, f5 ] )
gap> Source( Embedding( Source( info2.projection ), 1 ) ) = g2;
true

5.3-4 IdempotentEndomorphisms
‣ IdempotentEndomorphisms( G )( operation )
‣ IdempotentEndomorphismsData( G )( attribute )
‣ IdempotentEndomorphismsWithImage( genG, R )( operation )

An endomorphism \(f : G \to G\) is idempotent if \(f^2=f\). It has an image \(R \leqslant G\); is the identity map when restricted to \(R\); and has a kernel \(N\) which has trivial intersection with \(R\) and has size \(|G|/|R|\).

The operation IdempotentEndomorphismsWithImage(genG,R) returns a list of the images of the generating set genG of a group \(G\) under the idempotent endomorphisms with image \(R\).

The attribute IdempotentEndomorphismsData(G) returns a record data with fields data.gens, a fixed generating set for \(G\), and data.images a list of the non-empty outputs of IdempotentEndomorphismsWithImage(genG,R) obtained by iterating over all subgroups of \(G\).

The operation IdempotentEndomorphisms(G) returns the list of these mappings obtained using IdempotentEndomorphismsData(G). The first of these is the zero map, the second is the identity.


gap> gens := [ (1,2,3,4), (1,2)(3,4) ];; 
gap> d8 := Group( gens );;
gap> SetName( d8, "d8" );
gap> c2 := Subgroup( d8, [ (2,4) ] );;
gap> IdempotentEndomorphismsWithImage( gens, c2 );
[ [ (), (2,4) ], [ (2,4), () ] ]
gap> IdempotentEndomorphismsData( d8 );
rec( gens := [ (1,2,3,4), (1,2)(3,4) ], 
  images := [ [ [ (), () ] ], [ [ (), (2,4) ], [ (2,4), () ] ], 
      [ [ (), (1,3) ], [ (1,3), () ] ], 
      [ [ (), (1,2)(3,4) ], [ (1,2)(3,4), (1,2)(3,4) ] ], 
      [ [ (), (1,4)(2,3) ], [ (1,4)(2,3), (1,4)(2,3) ] ], 
      [ [ (1,2,3,4), (1,2)(3,4) ] ] ] )
gap> List( last.images, L -> Length(L) );
[ 1, 2, 2, 2, 2, 1 ]
gap> IdempotentEndomorphisms( d8 );               
[ [ (1,2,3,4), (1,2)(3,4) ] -> [ (), () ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (2,4) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (2,4), () ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (1,3) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,3), () ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (1,2)(3,4) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,2)(3,4), (1,2)(3,4) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (), (1,4)(2,3) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,4)(2,3), (1,4)(2,3) ], 
  [ (1,2,3,4), (1,2)(3,4) ] -> [ (1,2,3,4), (1,2)(3,4) ] ]

The quaternion group q8 is an example of a group with a tail: there is only one subgroup in the lattice which covers the identity subgroup. The only idempotent isomorphisms of such groups are the identity mapping and the zero mapping because the only pairs \(N,R\) are the whole group and the identity subgroup.


gap> q8 := QuaternionGroup( 8 );;
gap> IdempotentEndomorphisms( q8 );
[ [ x, y ] -> [ <identity> of ..., <identity> of ... ], [ x, y ] -> [ x, y ] ]

5.3-5 DirectProductOfFunctions
‣ DirectProductOfFunctions( G, H, f1, f2 )( operation )

Given group homomorphisms \(f_1 : G_1 \to G_2\) and \(f_2 : H_1 \to H_2\), this operation return the product homomorphism \(f_1 \times f_2 : G_1 \times G_2 \to H_1 \times H_2\).


gap> c4 := Group( (1,2,3,4) );; 
gap> c2 := Group( (5,6) );; 
gap> f1 := GroupHomomorphismByImages( c4, c2, [(1,2,3,4)], [(5,6)] );;
gap> c3 := Group( (1,2,3) );; 
gap> c6 := Group( (1,2,3,4,5,6) );; 
gap> f2 := GroupHomomorphismByImages( c3, c6, [(1,2,3)], [(1,3,5)(2,4,6)] );; 
gap> c4c3 := DirectProduct( c4, c3 ); 
Group([ (1,2,3,4), (5,6,7) ])
gap> c2c6 := DirectProduct( c2, c6 ); 
Group([ (1,2), (3,4,5,6,7,8) ])
gap> f := DirectProductOfFunctions( c4c3, c2c6, f1, f2 ); 
[ (1,2,3,4), (5,6,7) ] -> [ (1,2), (3,5,7)(4,6,8) ]
gap> ImageElm( f, (1,4,3,2)(5,7,6) ); 
(1,2)(3,7,5)(4,8,6)

5.3-6 DirectProductOfAutomorphismGroups
‣ DirectProductOfAutomorphismGroups( A1, A2 )( operation )

Let \(A_1,A_2\) be groups of automorphism of groups \(G_1,G_2\) respectively. The output of this function is a group \(A_1 \times A_2\) of automorphisms of \(G_1 \times G_2\).


gap> c9 := Group( (1,2,3,4,5,6,7,8,9) );; 
gap> ac9 := AutomorphismGroup( c9 );; 
gap> q8 := QuaternionGroup( IsPermGroup, 8 );;
gap> aq8 := AutomorphismGroup( q8 );;
gap> A := DirectProductOfAutomorphismGroups( ac9, aq8 );
<group with 5 generators>
gap> genA := GeneratorsOfGroup( A );;
gap> G := Source( genA[1] );
Group([ (1,2,3,4,5,6,7,8,9), (10,14,12,16)(11,17,13,15), (10,11,12,13)
(14,15,16,17) ])
gap> a := genA[1]*genA[5];  
[ (1,2,3,4,5,6,7,8,9), (10,14,12,16)(11,17,13,15), (10,11,12,13)(14,15,16,17) 
 ] -> [ (1,3,5,7,9,2,4,6,8), (10,16,12,14)(11,15,13,17), 
  (10,11,12,13)(14,15,16,17) ]
gap> ImageElm( a, (1,9,8,7,6,5,4,3,2)(10,14,12,16)(11,17,13,15) );
(1,8,6,4,2,9,7,5,3)(10,16,12,14)(11,15,13,17)
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utils-0.86/doc/toggless.js000644 000765 000024 00000004205 14621610515 020647 0ustar00christopherwensleystaff000000 000000 /* toggless.js Frank Lübeck */ /* this file contains two functions: mergeSideTOCHooks: this changes div.ContSect elements to the class ContSectClosed and includes a hook to toggle between ContSectClosed and ContSectOpen. openclosetoc: this function does the toggling, the rest is done by CSS */ closedTOCMarker = "▶ "; openTOCMarker = "▼ "; noTOCMarker = " "; /* merge hooks into side toc for opening/closing subsections with openclosetoc */ function mergeSideTOCHooks() { var hlist = document.getElementsByTagName("div"); for (var i = 0; i < hlist.length; i++) { if (hlist[i].className == "ContSect") { var chlds = hlist[i].childNodes; var el = document.createElement("span"); var oncl = document.createAttribute("class"); oncl.nodeValue = "toctoggle"; el.setAttributeNode(oncl); var cont; if (chlds.length > 2) { var oncl = document.createAttribute("onclick"); oncl.nodeValue = "openclosetoc(event)"; el.setAttributeNode(oncl); cont = document.createTextNode(closedTOCMarker); } else { cont = document.createTextNode(noTOCMarker); } el.appendChild(cont); hlist[i].firstChild.insertBefore(el, hlist[i].firstChild.firstChild); hlist[i].className = "ContSectClosed"; } } } function openclosetoc (event) { /* first two steps to make it work in most browsers */ var evt=window.event || event; if (!evt.target) evt.target=evt.srcElement; var markClosed = document.createTextNode(closedTOCMarker); var markOpen = document.createTextNode(openTOCMarker); var par = evt.target.parentNode.parentNode; if (par.className == "ContSectOpen") { par.className = "ContSectClosed"; evt.target.replaceChild(markClosed, evt.target.firstChild); } else if (par.className == "ContSectClosed") { par.className = "ContSectOpen"; evt.target.replaceChild(markOpen, evt.target.firstChild); } } /* adjust jscontent which is called onload */ jscontentfuncs.push(mergeSideTOCHooks); utils-0.86/doc/iterator.xml000644 000765 000024 00000012325 14621444241 021040 0ustar00christopherwensleystaff000000 000000 Iterators
Some iterators for groups and their isomorphisms Iterators The motivation for adding these operations is partly to give a simple example of an iterator for a list that does not yet exist, and need not be created.

The main &GAP; library contains functions producing complete lists of group homomorphisms such as AllHomomorphisms; AllEndomorphisms and AllAutomorphisms. Here we add the missing AllIsomorphisms(G,H) for a list of isomorphisms from G to H. The method is simple -- find one isomorphism G \to H and compose this with all the automorphisms of G. In all these cases it may not be desirable to construct a list of homomorphisms, but just implement an iterator, and that is what is done here. The operation AllIsomorphismsNumber returns the number of isomorphisms iterated over (this is, of course, just the order of the automorphisms group). The operation AllIsomorphisms produces the list or isomorphisms.

G := SmallGroup( 6,1);; gap> iter := AllIsomorphismsIterator( G, s3 );; gap> NextIterator( iter ); [ f1, f2 ] -> [ (6,7), (5,6,7) ] gap> n := AllIsomorphismsNumber( G, s3 ); 6 gap> AllIsomorphisms( G, s3 ); [ [ f1, f2 ] -> [ (6,7), (5,6,7) ], [ f1, f2 ] -> [ (5,7), (5,6,7) ], [ f1, f2 ] -> [ (5,6), (5,7,6) ], [ f1, f2 ] -> [ (6,7), (5,7,6) ], [ f1, f2 ] -> [ (5,7), (5,7,6) ], [ f1, f2 ] -> [ (5,6), (5,6,7) ] ] gap> iter := AllIsomorphismsIterator( G, s3 );; gap> for h in iter do Print( ImageElm( h, G.1 ) = (6,7), ", " ); od; true, false, false, true, false, false, ]]> The manual entry for the operation AllSubgroups states that it is only intended to be used on small examples in a classroom situation. Access to all subgroups was required by the XMod package, so this iterator was introduced here. It used the operations LatticeSubgroups(G) and ConjugacyClassesSubgroups(lat), and then iterates over the entries in these classes.

c3c3 := Group( (1,2,3), (4,5,6) );; gap> iter := AllSubgroupsIterator( c3c3 ); gap> while not IsDoneIterator(iter) do Print(NextIterator(iter),"\n"); od; Group( () ) Group( [ (4,5,6) ] ) Group( [ (1,2,3) ] ) Group( [ (1,2,3)(4,5,6) ] ) Group( [ (1,3,2)(4,5,6) ] ) Group( [ (4,5,6), (1,2,3) ] ) ]]>

Operations on iterators This section considers ways of producing an iterator from one or more iterators. It may be that operations equivalent to these are available elsewhere in the library -- if so, the ones here can be removed in due course. This iterator returns all pairs [x,y] where x is the output of a first iterator and y is the output of a second iterator.

it1 := Iterator( [ 1, 2, 3 ] );; gap> it2 := Iterator( [ 4, 5, 6 ] );; gap> iter := CartesianIterator( it1, it2 );; gap> while not IsDoneIterator(iter) do Print(NextIterator(iter),"\n"); od; [ 1, 4 ] [ 1, 5 ] [ 1, 6 ] [ 2, 4 ] [ 2, 5 ] [ 2, 6 ] [ 3, 4 ] [ 3, 5 ] [ 3, 6 ] ]]> This operation returns pairs [x,y] where x,y are output from a given iterator iter. Unlike the output from CartesianIterator(iter,iter), unordered pairs are returned. In the case L = [1,2,3,\ldots] the pairs are ordered as [1,1],[1,2],[2,2],[1,3],[2,3],[3,3],\ldots.

L := [6,7,8,9];; gap> iterL := IteratorList( L );; gap> pairsL := UnorderedPairsIterator( iterL );; gap> while not IsDoneIterator(pairsL) do Print(NextIterator(pairsL),"\n"); od; [ 6, 6 ] [ 6, 7 ] [ 7, 7 ] [ 6, 8 ] [ 7, 8 ] [ 8, 8 ] [ 6, 9 ] [ 7, 9 ] [ 8, 9 ] [ 9, 9 ] gap> iter4 := IteratorList( [ 4 ] ); gap> pairs4 := UnorderedPairsIterator(iter4); gap> NextIterator( pairs4 ); [ 4, 4 ] gap> IsDoneIterator( pairs4 ); true ]]>

utils-0.86/doc/chap7.txt000644 000765 000024 00000015723 14621610512 020231 0ustar00christopherwensleystaff000000 000000 7 Iterators 7.1 Some iterators for groups and their isomorphisms The motivation for adding these operations is partly to give a simple example of an iterator for a list that does not yet exist, and need not be created. 7.1-1 AllIsomorphismsIterator AllIsomorphismsIterator( G, H )  operation AllIsomorphismsNumber( G, H )  operation AllIsomorphisms( G, H )  operation The main GAP library contains functions producing complete lists of group homomorphisms such as AllHomomorphisms; AllEndomorphisms and AllAutomorphisms. Here we add the missing AllIsomorphisms(G,H) for a list of isomorphisms from G to H. The method is simple -- find one isomorphism G -> H and compose this with all the automorphisms of G. In all these cases it may not be desirable to construct a list of homomorphisms, but just implement an iterator, and that is what is done here. The operation AllIsomorphismsNumber returns the number of isomorphisms iterated over (this is, of course, just the order of the automorphisms group). The operation AllIsomorphisms produces the list or isomorphisms.  Example   gap> G := SmallGroup( 6,1);;  gap> iter := AllIsomorphismsIterator( G, s3 );; gap> NextIterator( iter ); [ f1, f2 ] -> [ (6,7), (5,6,7) ] gap> n := AllIsomorphismsNumber( G, s3 ); 6 gap> AllIsomorphisms( G, s3 ); [ [ f1, f2 ] -> [ (6,7), (5,6,7) ], [ f1, f2 ] -> [ (5,7), (5,6,7) ],   [ f1, f2 ] -> [ (5,6), (5,7,6) ], [ f1, f2 ] -> [ (6,7), (5,7,6) ],   [ f1, f2 ] -> [ (5,7), (5,7,6) ], [ f1, f2 ] -> [ (5,6), (5,6,7) ] ] gap> iter := AllIsomorphismsIterator( G, s3 );; gap> for h in iter do Print( ImageElm( h, G.1 ) = (6,7), ", " ); od; true, false, false, true, false, false,   7.1-2 AllSubgroupsIterator AllSubgroupsIterator( G )  operation The manual entry for the operation AllSubgroups states that it is only intended to be used on small examples in a classroom situation. Access to all subgroups was required by the XMod package, so this iterator was introduced here. It used the operations LatticeSubgroups(G) and ConjugacyClassesSubgroups(lat), and then iterates over the entries in these classes.  Example   gap> c3c3 := Group( (1,2,3), (4,5,6) );;  gap> iter := AllSubgroupsIterator( c3c3 );  gap> while not IsDoneIterator(iter) do Print(NextIterator(iter),"\n"); od; Group( () ) Group( [ (4,5,6) ] ) Group( [ (1,2,3) ] ) Group( [ (1,2,3)(4,5,6) ] ) Group( [ (1,3,2)(4,5,6) ] ) Group( [ (4,5,6), (1,2,3) ] )   7.2 Operations on iterators This section considers ways of producing an iterator from one or more iterators. It may be that operations equivalent to these are available elsewhere in the library -- if so, the ones here can be removed in due course. 7.2-1 CartesianIterator CartesianIterator( iter1, iter2 )  operation This iterator returns all pairs [x,y] where x is the output of a first iterator and y is the output of a second iterator.  Example   gap> it1 := Iterator( [ 1, 2, 3 ] );; gap> it2 := Iterator( [ 4, 5, 6 ] );; gap> iter := CartesianIterator( it1, it2 );; gap> while not IsDoneIterator(iter) do Print(NextIterator(iter),"\n"); od; [ 1, 4 ] [ 1, 5 ] [ 1, 6 ] [ 2, 4 ] [ 2, 5 ] [ 2, 6 ] [ 3, 4 ] [ 3, 5 ] [ 3, 6 ]   7.2-2 UnorderedPairsIterator UnorderedPairsIterator( iter )  operation This operation returns pairs [x,y] where x,y are output from a given iterator iter. Unlike the output from CartesianIterator(iter,iter), unordered pairs are returned. In the case L = [1,2,3,...] the pairs are ordered as [1,1],[1,2],[2,2],[1,3],[2,3],[3,3],....  Example   gap> L := [6,7,8,9];; gap> iterL := IteratorList( L );;  gap> pairsL := UnorderedPairsIterator( iterL );;  gap> while not IsDoneIterator(pairsL) do Print(NextIterator(pairsL),"\n"); od; [ 6, 6 ] [ 6, 7 ] [ 7, 7 ] [ 6, 8 ] [ 7, 8 ] [ 8, 8 ] [ 6, 9 ] [ 7, 9 ] [ 8, 9 ] [ 9, 9 ] gap> iter4 := IteratorList( [ 4 ] );  gap> pairs4 := UnorderedPairsIterator(iter4);  gap> NextIterator( pairs4 ); [ 4, 4 ] gap> IsDoneIterator( pairs4 ); true   utils-0.86/doc/chap6.txt000644 000765 000024 00000012276 14621610512 020230 0ustar00christopherwensleystaff000000 000000 6 Matrices 6.1 Some operations for matrices 6.1-1 DirectSumDecompositionMatrices DirectSumDecompositionMatrices( M )  operation In June 2023 Hongyi Zhao asked in the Forum for a function to implement matrix decomposition into blocks. Such a function was then provided by Pedro García-Sánchez. Hongyi Zhao then requested that the function be added to Utils. What is provided here is a revised version of the original solution, returning a list of decompositions. This function is a partial inverse to the undocumented library operation DirectSumMat. So if L is the list of diagonal decompositions of a matrix M then each entry in L is a list of matrices, and the direct sum of each of these lists is equal to the original M. In the following examples, M_6 is an obvious direct sum with 3 blocks. M_4 is an example with three decompositions, while M_8 = M_4 ⊕ M_4 has 16 decompositions (not listed).  Example   gap> M6 := [ [1,2,0,0,0,0], [3,4,0,0,0,0], [5,6,0,0,0,0],  >  [0,0,9,0,0,0], [0,0,0,1,2,3], [0,0,0,4,5,6] ];; gap> Display( M6 ); [ [ 1, 2, 0, 0, 0, 0 ],  [ 3, 4, 0, 0, 0, 0 ],  [ 5, 6, 0, 0, 0, 0 ],  [ 0, 0, 9, 0, 0, 0 ],  [ 0, 0, 0, 1, 2, 3 ],  [ 0, 0, 0, 4, 5, 6 ] ] gap> L6 := DirectSumDecompositionMatrices( M6 ); [ [ [ [ 1, 2 ], [ 3, 4 ], [ 5, 6 ] ], [ [ 9 ] ], [ [ 1, 2, 3 ], [ 4, 5, 6 ] ]   ] ]  gap> M4 := [ [0,3,0,0], [0,0,0,0], [0,0,0,0], [0,0,4,0] ];; gap> Display( M4 ); [ [ 0, 3, 0, 0 ],  [ 0, 0, 0, 0 ],  [ 0, 0, 0, 0 ],  [ 0, 0, 4, 0 ] ] gap> L4 := DirectSumDecompositionMatrices( M4 ); [ [ [ [ 0, 3 ] ], [ [ 0, 0 ], [ 0, 0 ], [ 4, 0 ] ] ],   [ [ [ 0, 3 ], [ 0, 0 ] ], [ [ 0, 0 ], [ 4, 0 ] ] ],   [ [ [ 0, 3 ], [ 0, 0 ], [ 0, 0 ] ], [ [ 4, 0 ] ] ] ] gap> for L in L4 do  >  A := DirectSumMat( L );;  >  if ( A = M4 ) then Print( "yes, A = M4\n" ); fi;  >  od; yes, A = M4 yes, A = M4 yes, A = M4  gap> M8 := DirectSumMat( M4, M4 );;  gap> Display( M8 ); [ [ 0, 3, 0, 0, 0, 0, 0, 0 ],  [ 0, 0, 0, 0, 0, 0, 0, 0 ],  [ 0, 0, 0, 0, 0, 0, 0, 0 ],  [ 0, 0, 4, 0, 0, 0, 0, 0 ],  [ 0, 0, 0, 0, 0, 3, 0, 0 ],  [ 0, 0, 0, 0, 0, 0, 0, 0 ],  [ 0, 0, 0, 0, 0, 0, 0, 0 ],  [ 0, 0, 0, 0, 0, 0, 4, 0 ] ] gap> L8 := DirectSumDecompositionMatrices( M8 );; gap> Length( L8 );  16   The current method does not, however, catch all possible decompositions. In the following example the matrix M_5 has its third row and third column extirely zero, and the only decomposition found has a [0] factor. There are clearly two 2-factor decompositions with a 2-by-3 and a 3-by-2 factor, but these are not found at present.  Example   gap> M5 := [ [1,2,0,0,0], [3,4,0,0,0], [0,0,0,0,0], >  [0,0,0,6,7], [0,0,0,8,9] ];; gap> Display(M5); [ [ 1, 2, 0, 0, 0 ],  [ 3, 4, 0, 0, 0 ],  [ 0, 0, 0, 0, 0 ],  [ 0, 0, 0, 6, 7 ],  [ 0, 0, 0, 8, 9 ] ] gap> L5 := DirectSumDecompositionMatrices( M5 );  [ [ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 0 ] ], [ [ 6, 7 ], [ 8, 9 ] ] ] ]   utils-0.86/doc/chapInd_mj.html000644 000765 000024 00000022111 14621610515 021400 0ustar00christopherwensleystaff000000 000000 GAP (Utils) - Index
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Index

AllIsomorphisms 7.1-1
AllIsomorphismsIterator 7.1-1
AllIsomorphismsNumber 7.1-1
AllProducts 4.1-2
AllSmoothIntegers 4.1-1 4.1-1
AllSubgroupsIterator 7.1-2
AssignGlobals 8.1-1
BlankFreeString 3.3-1
CartesianIterator 7.2-1
CentralProduct 5.3-3
CentralProductInfo 5.3-3
Comm 5.1-1
CommonRepresentatives 3.2-1
CommonTransversal 3.2-1
ConvertToMagmaInputString 10.3-1
CreateDirIfMissing 11.1
DifferencesList 3.1-1
DirectProductOfAutomorphismGroups 5.3-6
DirectProductOfFunctions 5.3-5
DirectSumDecompositionMatrices 6.1-1
distinct and common representatives 3.2
DistinctRepresentatives 3.2-1
Download 9.1-1
DownloadVerifyCertificate 9.1-2
EpimorphismByGenerators 5.3-1
ExponentOfPrime 11.3-2
FindMatchingFiles 11.1
Fitting series 5.1-5
FittingLength 5.1-5
FloatQuotientsList 3.1-2
GeneratorsAndInverses 5.1-4
GetSuffix 11.1
GitHub repository 1.
IdempotentEndomorphisms 5.3-4
IdempotentEndomorphismsData 5.3-4
IdempotentEndomorphismsWithImage 5.3-4
IntOrOnfinityToLaTeX 10.2-1
IsCommonTransversal 3.2-1
IsCommuting 5.1-2
Iterators 7.1
LaTeXStringFactorsInt 10.2-2
LeftCoset 5.2-1
ListOfPowers 5.1-3
Log2HTML 10.1-1
LowerFittingSeries 5.1-5
MatrixGroupToMagmaFormat 10.3-1
NextProbablyPrimeInt 4.1-4
OKtoReadFromUtils 12.
OptionRecordWithDefaults 8.2-1
PcGroupToMagmaFormat 10.3-1
PermGroupToMagmaFormat 10.3-1
PrimeNumbersIterator 4.1-5
PrintApplicableMethod 11.3-1
PrintOneItemPerLine 11.2
PrintSelection 2.1-1 2.1-1
Pullback 5.3-2
PullbackInfo 5.3-2
QuotientsList 3.1-2
RandomCombination 3.1-4
RestrictedPartitionsWithoutRepetitions 4.1-3
SearchCycle 3.1-3
SetIfMissing 11.1
smooth integer 4.1-1
StringDotSuffix 11.1
UnorderedPairsIterator 7.2-2
UpperFittingSeries 5.1-5

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12 The transfer procedure

12 The transfer procedure

We consider here the process for transferring utility functions from a package Home to Utils which has to avoid the potential problem of duplicate declarations of a function causing loading problems in GAP.

If the functions in Home all have names of the form HOME_FunctionName then, in Utils, these functions are likely to be renamed as FunctionName or something similar. In this case the problem of duplicate declarations does not arise. This is what has happened with transfers from the AutoDoc package.

The case where the function names are unchanged is more complicated. Initially we tried out a process which allowed repeated declarations and installations of the functions being transferred. This involved additions to the main library files global.g and oper.g. Since there were misgivings about interfering in this way with basic operations such as BIND_GLOBAL, a simpler (but slightly less convenient) process has been adopted.

Using this alternative procedure, the following steps will be followed when making transfers from Home to Utils.

  1. (Home:) Offer functions for inclusion. This may be simply done by emailing a list of functions. More usefully, email the declaration, implementation, test and documentation files, e.g.: home.gd, home.gi, home.tst and home.xml. (All active authors should be involved.)

  2. (Home:) Declare that m.n is the last version of Home to contain these functions, so that m.n+1 (or similar) will be the first version of Home to have all these functions removed, and to specify Utils as a required package.

  3. (Utils:) Add strings "home" and "m.n" to the list UtilsPackageVersions in the file utils/lib/start.gd.

    
UtilsPackageVersions := 
  [ "autodoc",     "2016.01.31", 
    "resclasses",  "4.2.5", 
    "home",        "m.n",
    ...,           ... 
  ];

    While the transfers are being made, it is essential that any new versions of Home should be tested with the latest version of Utils before they are released, so as to avoid loading failures.

  4. (Utils:) Include the function declaration and implementation sections in suitable files, enclosed within a conditional clause of the form:

    
if OKtoReadFromUtils( "Home" ) then
. . . . . . 
 <the code> 
. . . . . . 
fi;

    The function OKtoReadFromUtils returns true only if there is an installed version of Home and if this version is greater than m.n. So, at this stage, the copied code will not be read, and the transferred functions can only be called if Home has been installed.

  5. (Utils:) Add the test and documentation material to the appropriate files. The copied code can be tested by temporarily moving Home away from GAP's package directory.

  6. (Utils:) Release a new version of Utils containing all the transferred material.

  7. (Home:) Edit out the declarations and implementations of all the transferred functions, and remove references to them in the manual and tests. Possibly add a note to the manual that these functions have been transferred. Add Utils to the list of Home's required packages in PackageInfo.g. Release a new version of Home.

  8. (Utils:) In due course, when the new version(s) of Home are well established, it may be safe to remove the conditional clauses mentioned in item 4 above. The entry for Home in UtilsPackageLists may then be removed.

Finally, a note on the procedure for testing these functions. As long as a function being transferred still exists in the Home package, the code will not be read from Utils. So, when the tests are run, it is necessary to LoadPackage("home") before the function is called. The file utils/tst/testall.g makes sure that all the necessary packages are loaded before the individual tests are called.

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utils-0.86/lib/print.gd000644 000765 000024 00000001644 14621444241 020140 0ustar00christopherwensleystaff000000 000000 ############################################################################## ## #W print.gd GAP4 package `Utils' Chris Wensley ## #Y Copyright (C) 2015-2018, The GAP Group ############################################################################## ## added global function and methods to print subsets of lists/iterators ## DeclareGlobalFunction( "PrintSelection" ); DeclareOperation( "PrintSelectionFromList", [ IsList, IsPosInt, IsPosInt, IsPosInt ] ); DeclareOperation( "PrintSelectionFromIterator", [ IsIterator, IsPosInt, IsPosInt, IsAdditiveElement ] ); DeclareOperation( "PrintSelectionFromListByList", [ IsList, IsList ] ); DeclareOperation( "PrintSelectionFromIteratorByList", [ IsIterator, IsList ] ); ############################################################################# ## #E print.gd . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/lcset.gi000644 000765 000024 00000011271 14621444241 020120 0ustar00christopherwensleystaff000000 000000 ############################################################################## ## #W lcset.gi GAP4 package `Utils' Chris Wensley ## #Y Copyright (C) 2015-2022, The GAP Group ############################################################################# ## #O LeftCoset( , ) . . . . . . . . forms a left coset of U by g \in G ## InstallMethod( LeftCoset, "generic method for left cosets", true, [ IsObject, IsGroup ], 0, function( g, U ) local lc, fam; fam := FamilyObj( U ); if not IsBound( fam!.leftCosetsDefaultType ) then fam!.leftCosetsDefaultType := NewType( fam, IsLeftCosetDefaultRep and HasActingDomain and HasFunctionAction and HasRepresentative ); fi; lc := rec(); ObjectifyWithAttributes( lc, fam!.leftCosetsDefaultType, Representative, g, ActingDomain, U, FunctionAction, OnLeftInverse, IsLeftCoset, true ); if HasSize( U ) then SetSize( lc, Size( U ) ); fi; return lc; end); ############################################################################# ## #A Inverse( ) . . . . . forms the right coset Ug^-1 corresponding to gU #A Inverse( ) . . . . . forms the left coset g^-1U corresponding to Ug ## InstallOtherMethod( Inverse, "generic method for left cosets", true, [ IsLeftCoset ], 0, function( lc ) local g, U, rc; g := Representative( lc ); U := ActingDomain( lc ); rc := RightCoset( U, g^-1 ); SetInverse( rc, lc ); return rc; end); InstallOtherMethod( Inverse, "generic method for right cosets", true, [ IsRightCoset ], 0, function( rc ) local g, U, lc; g := Representative( rc ); U := ActingDomain( rc ); lc := LeftCoset( g^-1, U ); SetInverse( lc, rc ); return lc; end); InstallMethod( PrintString, "for a left coset", true, [ IsLeftCoset ], 0, function( d ) return STRINGIFY( "LeftCoset(\<", PrintString( Representative(d) ), ",\>", PrintString( ActingDomain(d) ), ")" ); end); InstallMethod( PrintObj, "for a left coset", true, [ IsLeftCoset ], 0, function( d ) Print( PrintString( d ) ); end); InstallMethod( ViewString, "for a left coset", true, [ IsLeftCoset ], 0, function( d ) return STRINGIFY( "LeftCoset(\<", ViewString( Representative(d) ), ",\>", ViewString( ActingDomain(d) ), ")" ); end); InstallMethod( ViewObj, "for a left coset", true, [ IsLeftCoset ], 0, function( d ) Print( ViewString( d ) ); end); InstallMethod( \=, "for two left cosets", IsIdenticalObj, [ IsLeftCoset, IsLeftCoset ], 0, function( lc1, lc2 ) return ( ActingDomain( lc1 ) = ActingDomain( lc2 ) ) and ( Representative(lc1)/Representative(lc2) in ActingDomain( lc1 ) ); end); InstallMethod( \in, "element and LeftCoset", true, [ IsMultiplicativeElementWithInverse, IsLeftCoset ], 100, function( g, lc ) return ( Representative(lc)^-1 * g ) in ActingDomain( lc ); end); InstallOtherMethod( \*, "element and LeftCoset", true, [ IsMultiplicativeElementWithInverse, IsLeftCoset ], 0, function( g, lc ) return LeftCoset( g * Representative( lc ), ActingDomain( lc ) ); end); InstallOtherMethod( \^, "LeftCoset and element", true, [ IsLeftCoset, IsMultiplicativeElementWithInverse ], 0, function( lc, g ) return g^-1 * lc; end); InstallMethod( Size, "for a left coset", true, [ IsLeftCoset ], 0, function( lc ) return Size( ActingDomain( lc ) ); end); InstallOtherMethod( IsBiCoset, "for a left coset", true, [ IsLeftCoset ], 0, function( lc ) return IsBiCoset( Inverse( lc ) ); end); InstallOtherMethod( AsSet, "for a left coset", true, [ IsLeftCoset ], 0, function( lc ) local L; L := AsSet( Inverse( lc ) ); return Set( L, g -> g^-1 ); end); InstallOtherMethod( Intersection2, "for two left cosets", true, [ IsLeftCoset, IsLeftCoset ], 0, function( lc1, lc2 ) local L; L := Intersection2( Inverse( lc1 ), Inverse( lc2 ) ); return Set( L, g -> g^-1 ); ## return Inverse( L ); to be used once 4r12 is out end); ############################################################################# ## #F LeftCosets( ) . . . . . . . . . . . . left cosets of U by g \in G #O LeftCosetsNC( ) . . . . . . . . . . . left cosets of U by g \in G ## InstallGlobalFunction( LeftCosets, function( G, U ) if not IsSubset( G, U ) then Error("not contained"); fi; return LeftCosetsNC( G, U ); end ); InstallMethod( LeftCosetsNC, "generic", IsIdenticalObj, [ IsGroup, IsGroup ], 0, function( G, U ) local L; L := RightCosetsNC( G, U ); return List( L, Inverse ); end); utils-0.86/lib/lists.gi000644 000765 000024 00000023217 14621444241 020147 0ustar00christopherwensleystaff000000 000000 ############################################################################# ## #W lists.gi GAP4 package `Utils' Stefan Kohl ## #Y Copyright (C) 2015-2018, The GAP Group ############################################################################# ## these functions have been transferred from ResClasses ## #F DifferencesList( ) . . . . differences of consecutive list entries #F QuotientsList( ) . . . . . . quotients of consecutive list entries #F FloatQuotientsList( ) . . . . . . . . . . . . dito, but as floats ## InstallGlobalFunction( DifferencesList, list -> List( [ 2..Length(list) ], pos -> list[ pos ] - list[ pos-1 ] ) ); InstallGlobalFunction( QuotientsList, function( list ) local len, pos, quot; len := Length( list ); quot := ListWithIdenticalEntries( len-1, 0 ); for pos in [1..len-1] do if IsZero( list[pos] ) then quot[pos] := fail; else quot[pos] := list[pos+1]/list[pos]; fi; od; return quot; end ); InstallGlobalFunction( FloatQuotientsList, list -> List( QuotientsList( list ), Float ) ); ############################################################################# ## this function has been transferred from ResClasses ## #M RandomCombination( S, k ) . . . . . . . . . . . . . . . . default method ## InstallMethod( RandomCombination, "default method", ReturnTrue, [ IsListOrCollection, IsPosInt ], function ( S, k ) local c, elm, i; if k > Size(S) then return fail; fi; c := []; for i in [1..k] do repeat elm := Random(S); until not elm in c; Add(c,elm); od; return Set(c); end ); ############################################################################# ## this function has been transferred from RCWA ## #F SearchCycle( ) . a utility function for detecting cycles in lists ## InstallGlobalFunction( SearchCycle, function ( list ) local preperiod, cycle, startpos, mainpart, mainpartdiffs, elms, inds, min, n, d, i, j; n := Length(list); mainpart := list{[Int(n/3)..n]}; elms := Set(mainpart); cycle := [elms[1]]; startpos := Filtered(Positions(list,elms[1]),i->i>n/3); if Length(elms) = 1 then if ValueOption("alsopreperiod") <> true then return cycle; else i := Length(list); repeat i := i - 1; until i = 0 or list[i] <> elms[1]; preperiod := list{[1..i]}; return [preperiod,cycle]; fi; fi; i := 0; repeat i := i + 1; inds := Intersection(startpos+i,[1..n]); if inds = [] then return fail; fi; min := Minimum(list{inds}); Add(cycle,min); startpos := Filtered(startpos,j->j+i<=n and list[j+i]=min); if Length(startpos) <= 1 then return fail; fi; mainpartdiffs := DifferencesList(Intersection(startpos,[Int(n/3)..n])); if mainpartdiffs = [] then return fail; fi; d := Maximum(mainpartdiffs); until Length(cycle) = d; if Minimum(startpos) > n/2 or n-Maximum(startpos)-d+1 > d or list{[Maximum(startpos)+d..n]}<>cycle{[1..n-Maximum(startpos)-d+1]} then return fail; fi; if ValueOption("alsopreperiod") <> true then return cycle; else i := Minimum(startpos) + Length(cycle); repeat i := i - Length(cycle); until i <= 0 or list{[i..i+Length(cycle)-1]} <> cycle; preperiod := list{[1..i+Length(cycle)-1]}; return [preperiod,cycle]; fi; end ); ############################################################################## ## this function has been transferred from XMod ## #M DistinctRepresentatives( ) ## InstallMethod( DistinctRepresentatives, "for a list of sets", true, [ IsList ], 0, function( L ) local n, rep, U, len, i, j, k, used, found, S, T, M, P, x, y, z; if not ( IsList( L ) and ( ForAll( L, IsList ) or ForAll( L, IsSet ) ) ) then Error( "argument should be a list of sets" ); fi; n := Length( L ); U := [1..n]; len := 0 * U; for i in U do S := L[i]; if IsList( S ) then S := Set( S ); fi; len[i] := Length( S ); if ( len[i] = 0 ) then Error( "subsets must be non-empty" ); fi; if not ForAll( S, j -> ( j in U ) ) then Error( "each set must be a subset of [1..n]" ); fi; od; rep := 0 * U; used := 0 * U; rep[1] := L[1][1]; used[ rep[1] ] := 1; for i in [2..n] do found := false; S := L[i]; j := 0; while ( ( j < len[i] ) and not found ) do j := j+1; x := S[j]; if ( used[x] = 0 ) then rep[i] := x; used[x] := i; found := true; fi; od; # construct the graph component T := ShallowCopy( S ); M := List( T ); P := 0 * U; for x in M do P[x] := i; od; j := 0; while not found do j := j+1; x := M[j]; k := used[x]; if ( k = 0 ) then # reassign representatives y := P[x]; while ( y <> i ) do z := rep[y]; rep[y] := x; used[x] := y; x := z; y := P[x]; od; rep[i] := x; used[x] := i; found := true; else for y in L[k] do if not ( y in T ) then Add( M, y ); P[y] := k; T := Union( T, [y] ); fi; od; fi; if ( ( not found ) and ( j = Length( M ) ) ) then Print( "Hall condition not satisfied!\n" ); return false; fi; od; od; return rep; end ); ############################################################################## ## this function has been transferred from XMod ## #M CommonRepresentatives( , ) ## InstallMethod( CommonRepresentatives, "for a pair of lists/sets", true, [ IsList, IsList ], 0, function( J, K ) local U, i, j, k, m, n, lenJ, lenK, S, L, I, rep, perm, common; if not ( ForAll( J, IsList ) or ForAll( J, IsSet ) ) then Error( "first argument should be a list of sets" ); fi; m := Length( J ); if not ( ForAll( K, IsList ) or ForAll( K, IsSet ) ) then Error( "second argument should be a list of sets" ); fi; n := Length( K ); if not ( m = n ) then Error( "lists and have unequal length" ); fi; U := [1..n]; lenJ := 0 * U; lenK := 0 * U; for i in U do S := J[i]; if IsList( S ) then S := Set( S ); fi; lenJ[i] := Length( S ); if ( lenJ[i] = 0 ) then Error( "sets must be non-empty" ); fi; S := K[i]; if IsList( S ) then S := Set( S ); fi; lenK[i] := Length( S ); if ( lenK[i] = 0 ) then Error( "sets must be non-empty" ); fi; od; L := List( U, x -> [ ] ); for i in U do S := J[i]; for j in U do I := Intersection( S, K[j] ); if ( Length( I ) > 0 ) then Add( L[i], j ); fi; od; od; rep := DistinctRepresentatives( L ); perm := PermList( rep ); K := Permuted( K, perm^-1 ); common := 0 * U; for i in U do I := Intersection( J[i], K[i] ); common[i] := I[1]; od; return [ common, rep ]; end ); ############################################################################## ## this function has been transferred from XMod ## #M CommonTransversal ## InstallMethod( CommonTransversal, "for left and right cosets of a subgroup", true, [ IsGroup, IsGroup ], 0, function( G, H ) local R, ER, EL, T; if not IsSubgroup( G, H ) then Error( " must be a subgroup of " ); fi; R := RightCosets( G, H ); ER := List( R, Elements ); EL := List( ER, C -> List( C, Inverse ) ); Info( InfoUtils, 3, "right cosets: ", ER ); Info( InfoUtils, 3, " left cosets: ", EL ); T := CommonRepresentatives( EL, ER ); return T[1]; end ); ############################################################################## ## this function has been transferred from XMod ## #M IsCommonTransversal ## InstallMethod( IsCommonTransversal, "for group, subgroup, list", true, [ IsGroup, IsGroup, IsList ], 0, function( G, H, T ) local eG, eH, oG, oH, g, h, t, pos, ind, found; if not IsSubgroup( G, H ) then Print( "second group must be subgroup of first\n" ); return fail; fi; oG := Size( G ); oH := Size( H ); eG := Elements( G ); eH := Elements( H ); ind := oG/oH; found := 0 * [1..oG]; for t in T do if not ( t in eG ) then Print( "element of T not in G\n" ); return false; fi; for h in eH do g := t*h; pos := Position( eG, g ); found[pos] := found[pos] +1; g := h*t; pos := Position( eG, g ); found[pos] := found[pos] + 1; od; od; for t in [1..oG] do if not ( found[t] = 2 ) then Print( eG[t], " found ", found[t], " times\n" ); return false; fi; od; return true; end ); ############################################################################# ## #E lists.gi . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/record.gd000644 000765 000024 00000002276 14621444241 020264 0ustar00christopherwensleystaff000000 000000 ############################################################################## ## #W record.gd GAP4 package `Utils' Sebastian Gutsche ## Max Horn ## Stefan Kohl #Y Copyright (C) 2015-2022, The GAP Group ############################################################################# ## this function has been transferred from RCWA ## #F AssignGlobals( ) ## ## This auxiliary function assigns the record components of to ## global variables with the same names. ## DeclareGlobalFunction( "AssignGlobals" ); ############################################################################# ## this function has been contributed by Christopher Jefferson ## #F OptionRecordWithDefaults( arg1, arg2, ..., ) ## ## This auxiliary function assigns the record components of to ## global variables with the same names. ## DeclareGlobalFunction( "OptionRecordWithDefaults" ); ############################################################################# ## #E record.gd . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/matrix.gi000644 000765 000024 00000017051 14621444241 020314 0ustar00christopherwensleystaff000000 000000 ############################################################################# ## #W matrix.gi GAP4 package `Utils' Pedro A. Garcia-Sanchez ## #Y Copyright (C) 2023, The GAP Group InstallMethod( DirectSumBlock, "for a matrix", [ IsMatrixOrMatrixObj, IsInt, IsList, IsList ], 0, function( M, level, blocks, answer ) local nrM, ncM, nM, nr0, nc0, found, nlzr, nlzc, ntzr, ntzc, nlz, ntz, B, nrB, ncB, nB, nr, nc, i, j, k, i0, j0, nzr, nzc, rest, sub, bls, nrbls, ncbls, pr1, diag0, up0, low0, imax, jmax, len1, ok, ans; nrM := NumberRows( M ); ncM := NumberColumns( M ); nM := Minimum( [ nrM, ncM ] ); ## deal with the diagonal case first if IsDiagonalMatrix( M ) then bls := List( [1..nM], i -> [ [ M[i,i] ] ] ); if ( ncM > nM ) then sub := ExtractSubMatrix( M, [nM], [nM..ncM] )[1]; bls[nM] := [ sub ]; fi; if ( nrM > nM ) then sub := ExtractSubMatrix( M, [nM..nrM], [nM] )[1]; bls[ nM] := [ sub ]; fi; blocks[1] := Concatenation( blocks[1], bls ); ans := Concatenation( blocks[1], blocks[2] ); if not ( ans in answer ) then Add( answer, ans ); fi; return 0; fi; ## check for leading or trailing rows or columns of zeros nlzr := 0; found := false; i := 0; while not found do i := i+1; if ForAll( [1..ncM], j -> M[i,j] = 0 ) then nlzr := nlzr + 1; else found := true; fi; od; nlzc := 0; found := false; j := 0; while not found do j := j+1; if ForAll( [1..nrM], i -> M[i,j] = 0 ) then nlzc := nlzc + 1; else found := true; fi; od; ntzr := 0; found := false; i := nrM + 1; while not found do i := i-1; if ForAll( [1..ncM], j -> M[i,j] = 0 ) then ntzr := ntzr + 1; else found := true; fi; od; ntzc := 0; found := false; j := ncM + 1; while not found do j := j-1; if ForAll( [1..nrM], i -> M[i,j] = 0 ) then ntzc := ntzc + 1; else found := true; fi; od; ## add any leading or trailing blocks [ [0] ] nlz := Minimum( [ nlzr, nlzc ] ); if ( nlz > 0 ) then blocks[1] := Concatenation( blocks[1], ListWithIdenticalEntries( nlz, [ [0] ] ) ); fi; ntz := Minimum( [ ntzr, ntzc ] ); if ( ntz > 0 ) then blocks[2] := Concatenation( ListWithIdenticalEntries( ntz, [ [0] ] ), blocks[2] ); fi; B := ExtractSubMatrix( M, [nlz+1..nrM-ntz], [nlz+1..ncM-ntz] ); nrB := NumberRows( B ); ncB := NumberColumns( B ); nB := Minimum( [nrB, ncB] ); bls := [ ]; if ( nrB = 1 ) or ( ncB = 1 ) then Add( blocks[1], B ); ans := Concatenation( blocks[1], blocks[2] ); if not ( ans in answer ) then Add( answer, ans ); fi; return 0; fi; ## find first non-zero entry in column 1 found := false; pr1 := 0; while ( not found ) and ( pr1 < nrB-1 ) do pr1 := pr1 + 1; found := B[pr1,1] <> 0; od; ## determine ranges for i0 and j0 diag0 := nB+1; i := nB+1; while ( i > 1 ) do i := i-1; if ( B[i,i] <> 0 ) then diag0 := i; fi; od; up0 := [diag0,diag0]; j := 1; found := false; while ( not found ) and ( j < ncB ) do j := j+1; k := Minimum( [ j-1, nrB ] ); for i in [1..k] do if ( B[i,j] <> 0 ) then up0 := [i,j]; found := true; fi; od; od; low0 := [diag0,diag0]; i := 1; found := false; while ( not found ) and ( i < nrB ) do i := i+1; k := Minimum( [ i-1, ncB ] ); for j in [1..k] do if ( B[i,j] <> 0 ) then low0 := [i,j]; found := true; fi; od; od; imax := diag0; jmax := diag0; if ( up0[2] < low0[2] ) then jmax := low0[2] - 1; fi; if ( low0[1] < up0[1] ) then imax := up0[1] - 1; fi; if ( up0[1] < low0[1] ) then imax := Minimum( imax, low0[1] - 1 ); fi; if ( low0[2] < up0[2] ) then jmax := Minimum( jmax, up0[2] - 1 ); fi; if ( low0[1] > nrB ) then imax := up0[1]; fi; if ( up0[1] > ncM ) then jmax := low0[2]; fi; i0 := 0; while ( i0 < imax ) do i0 := i0 + 1; i := i0; j0 := 0; while ( j0 < jmax ) do found := false; j0 := j0 + 1; if ( j0 <= pr1 ) then j := j0; rest := MutableCopyMatrix( B ); nr := NumberRows( rest ); nc := NumberColumns( rest ); i := i0; j := j0; nzr := true; nzc := true; while nzr or nzc do nzr := ForAny( [1..i], i1 -> ForAny( [j+1..nc], j1->rest[i1][j1]<>0 ) ); nzc := ForAny( [i+1..nr], i1 -> ForAny( [1..j], j1->rest[i1][j1]<>0 ) ); if nzr then j := j+1; fi; if nzc then i := i+1; fi; od; bls := List( [1..i], i1 -> rest[i1]{[1..j]} ); ## check if we have filled all columns and rows nzc := ntzc - ntz; nrbls := NumberRows( bls ); ncbls := NumberColumns( bls ); if ( nzc > 0 ) and ( ncbls + nzc = ncB ) then ## add zero columns at the end of the block j := j + nzc; bls := List( [1..i], i1 -> rest[i1]{[1..j]} ); fi; nzr := ntzr - ntz; if ( nzr > 0 ) and ( nrbls + nzr = nrB ) then ## add zero rows at the end of the block i := i + nzr; bls := List( [1..i], i1 -> rest[i1]{[1..j]} ); fi; Add( blocks[1], bls ); len1 := Length( blocks[1] ); rest := List( [i+1..nr], i1 -> rest[i1]{[j+1..nc]} ); if ( rest <> [ ] ) then ok := DirectSumBlock( rest,level+1,blocks,answer ); else ans := Concatenation( blocks[1], blocks[2] ); if not ( ans in answer ) then Add( answer, ans ); fi; fi; blocks := [ blocks[1]{[1..len1-1]}, blocks[2] ]; fi; od; od; return Length( answer ); end ); InstallMethod( DirectSumDecompositionMatrices, "for a matrix", [ IsMatrixOrMatrixObj ], 0, function( M ) local L, blocks, answer, a, ma; blocks := [ [], [] ]; answer := [ ]; L := DirectSumBlock( M, 1, blocks, answer ); ## check that the answers are correct for a in answer do ma := DirectSumMat( a ); if not ( ma = M ) then Display( ma ); Error( "direct sum of decomposition <> original matrix" ); fi; od; return answer; end ); utils-0.86/lib/maps.gi000644 000765 000024 00000021132 14621444241 017743 0ustar00christopherwensleystaff000000 000000 ############################################################################# ## #W maps.gi GAP4 package `Utils' Stefan Kohl ## Chris Wensley #Y Copyright (C) 2015-2019, The GAP Group ############################################################################# ## #F EpimorphismByGenerators( , ) . epi: gen's of ->gen's of ## InstallMethod( EpimorphismByGenerators, "for groups", ReturnTrue, [ IsGroup, IsGroup ], 0, function( G, H ) if not ( IsFreeGroup( G ) ) then Print( "Warning: calling GroupHomomorphismByImagesNC without checks\n" ); fi; return GroupHomomorphismByImagesNC( G, H, GeneratorsOfGroup(G), GeneratorsOfGroup(H) ); end ); ############################################################################## ## #M Pullback . . . . . . . . . for two group homomorphisms with a common range ## InstallMethod( Pullback, "for two group homomorphisms", true, [ IsGroupHomomorphism, IsGroupHomomorphism ], 0, function( nu, mu ) local M, N, P, NxM, projM, pmu, projN, pnu, genNxM, e, genL, L, imphi, phi, impsi, psi, info; M := Source( mu ); P := Range( mu ); N := Source( nu ); if not ( Range(nu) = P ) then Error( "homs nu,mu should have a common range" ); fi; NxM := DirectProduct( N, M ); genNxM := GeneratorsOfGroup( NxM ); projN := Projection( NxM, 1 ); pnu := projN * nu; projM := Projection( NxM, 2 ); pmu := projM * mu; if IsFinite( NxM ) then genL := [ ]; L := Subgroup( NxM, [ One(NxM) ] ); for e in NxM do if ImageElm( pnu, e ) = ImageElm( pmu, e ) then if not ( e in L ) then Add( genL, e ); L := Group( genL ); fi; fi; od; else return fail; fi; imphi := List( genL, g -> ImageElm( projN, g ) ); phi := GroupHomomorphismByImages( L, N, genL, imphi ); impsi := List( genL, g -> ImageElm( projM, g ) ); psi := GroupHomomorphismByImages( L, M, genL, impsi ); info := rec( directProduct := NxM, projections := [phi,psi] ); SetPullbackInfo( L, info ); return L; end ); ############################################################################# ## #M CentralProduct( , , , ) ## InstallMethod( CentralProduct, [ "IsGroup", "IsGroup", "IsGroup", "IsGroupHomomorphism" ], function( G1, G2, Z1, Phi ) local gens, imgs, dp, emb1, emb2, N, proj, G; if not ( IsSubset( G1, Z1 ) and IsCentral( G1, Z1 ) ) then Error( " must be a central subgroup of " ); fi; gens:= GeneratorsOfGroup( Z1 ); imgs:= List( gens, x -> (x^-1)^Phi ); if not ( IsSubset( G2, imgs ) and ForAll( imgs, x -> IsCentral( G2, x ) ) ) then Error( " must map to a central subgroup of " ); fi; dp:= DirectProduct( G1, G2 ); emb1:= Embedding( dp, 1 ); emb2:= Embedding( dp, 2 ); N:= SubgroupNC( dp, List( [ 1 .. Length( gens ) ], i -> gens[i]^emb1 * imgs[i]^emb2 ) ); proj:= NaturalHomomorphismByNormalSubgroup( dp, N ); G:= Image( proj ); SetCentralProductInfo( G, rec( projection:= proj, phi:= Phi ) ); return G; end ); ############################################################################## ## #M IdempotentEndomorphisms . . . . . . . . . . . . . . . . . . . for a group #M IdempotentEndomorphismsData . . . . . . . . . . . . . . . . . for a group #M IdempotentEndomorphismsWithImage . . . . . for a group and a chosen image ## InstallMethod( IdempotentEndomorphismsWithImage, "for a list of group generators and a chosen image", [ IsList, IsGroup ], 0, function( genG, R ) local G, numG, r, q, norm, n, reps, i, j, rc; G := Group( genG ); if not IsSubgroup( G, R ) then Error( "R should be a subgroup of G" ); fi; numG := Length( genG ); r := Size( R ); q := Size( G )/r; norm := Filtered( NormalSubgroups( G ), N -> ( Size( N ) = q ) and IsTrivial( Intersection( N, R ) ) ); n := Length( norm ); reps := [ ]; for i in [1..n] do Add( reps, [ ] ); for j in [1..numG] do rc := norm[i]*genG[j]; Add( reps[i], First( rc, g -> g in R ) ); od; od; return reps; end ); InstallMethod( IdempotentEndomorphismsData, "for a group", [ IsGroup ], 0, function( G ) local genG, R, data, images; genG := SmallGeneratingSet( G ); images := [ ]; for R in AllSubgroups( G ) do data := IdempotentEndomorphismsWithImage( genG, R ); if ( data <> [ ] ) then Add( images, data ); fi; od; return rec( gens := genG, images := images ); end ); InstallMethod( IdempotentEndomorphisms, "for a group", [ IsGroup ], 0, function( G ) local data, genG, images, len, L, i, im; data := IdempotentEndomorphismsData( G ); genG := data!.gens; G := Group( genG ); images := data!.images; len := Length( images ); L := [ ]; for i in [1..len] do for im in images[i] do Add( L, GroupHomomorphismByImages( G, G, genG, im ) ); od; od; return L; end ); ############################################################################## ## #M DirectProductOfFunctions . . . . . . for two groups and two homomorphisms ## InstallMethod( DirectProductOfFunctions, "for two groups and two homs", [ IsGroup, IsGroup, IsGroupHomomorphism, IsGroupHomomorphism ], 0, function( G, H, f1, f2 ) local infoG, gpsG, G1, G2, infoH, gpsH, H1, H2, eG1, eG2, eH1, eH2, mgi1, mgi2, genG, imH1, imH2, imH; if not HasDirectProductInfo( G ) and HasDirectProductInfo( H ) then Error( "first two parameters should be direct products" ); fi; infoG := DirectProductInfo( G ); gpsG := infoG!.groups; G1 := gpsG[1]; G2 := gpsG[2]; if not ( ( G1 = Source( f1 ) ) and ( G2 = Source( f2 ) ) ) then Error( "f1,f2 should have source G1,G2" ); fi; eG1 := Embedding( G, 1 ); eG2 := Embedding( G, 2 ); mgi1 := MappingGeneratorsImages( f1 ); mgi2 := MappingGeneratorsImages( f2 ); infoH := DirectProductInfo( H ); gpsH := infoH!.groups; H1 := gpsH[1]; H2 := gpsH[2]; if not ( ( H1 = Range( f1 ) ) and ( H2 = Range( f2 ) ) ) then Error( "f1,f2 should have range H1,H2" ); fi; eH1 := Embedding( H, 1 ); eH2 := Embedding( H, 2 ); genG := Concatenation( List( mgi1[1], g -> ImageElm( eG1, g ) ), List( mgi2[1], g -> ImageElm( eG2, g ) ) ); imH1 := List( mgi1[2], h -> ImageElm( eH1, h ) ); imH2 := List( mgi2[2], h -> ImageElm( eH2, h ) ); imH := Concatenation( imH1, imH2 ); return GroupHomomorphismByImagesNC( G, H, genG, imH ); end ); ############################################################################## ## #M DirectProductOfAutomorphismGroups . . . . . . for two automorphism groups ## InstallMethod( DirectProductOfAutomorphismGroups, "for two groups", [ IsGroup, IsGroup ], 0, function( A1, A2 ) local gen1, gen2, G1, G2, id1, id2, dp, dp1, dp2, gen12, A12, em1, em2, pr1, pr2, info; if not ( IsGroupOfAutomorphisms(A1) and IsGroupOfAutomorphisms(A2) ) then Error( "A1,A2 should be automorphism groups" ); fi; gen1 := GeneratorsOfGroup( A1 ); gen2 := GeneratorsOfGroup( A2 ); G1 := Source( gen1[1] ); G2 := Source( gen2[1] ); if not ( IsGroup( G1 ) and IsGroup( G2 ) ) then Error( "A1,A2 should be automorphism groups of groups G1,G2" ); fi; id1 := IdentityMapping( G1 ); id2 := IdentityMapping( G2 ); dp := DirectProduct( G1, G2 ); dp1 := List( gen1, g -> DirectProductOfFunctions( dp, dp, g, id2 ) ); dp2 := List( gen2, g -> DirectProductOfFunctions( dp, dp, id1, g ) ); gen12 := Concatenation( dp1, dp2 ); A12 := Group( gen12 ); em1 := GroupHomomorphismByImages( A1, A12, gen1, dp1 ); em2 := GroupHomomorphismByImages( A2, A12, gen2, dp2 ); pr1 := GroupHomomorphismByImages( A12, A1, gen12, Concatenation( gen1, List( gen2, g -> id1 ) ) ); pr2 := GroupHomomorphismByImages( A12, A2, gen12, Concatenation( List( gen1, g -> id2 ), gen2 ) ); info := rec( embeddings := [ em1, em2 ], groups := [ A1, A2 ], projections := [ pr1, pr2 ] ); SetDirectProductInfo( A12, info ); return A12; end ); utils-0.86/lib/iterator.gd000644 000765 000024 00000003300 14621444241 020624 0ustar00christopherwensleystaff000000 000000 ############################################################################## ## #W iterator.gd GAP4 package `Utils' Chris Wensley ## #Y Copyright (C) 2015-2019, The GAP Group ############################################################################# ## These functions complement AllHomomorphisms, AllEndomorphisms, and ## AllAutomorphisms in the main library. ## The code gives an example of an iterator preferrred to a list. ## #O AllIsomorphismsIterator( , ) #F DoAllIsomorphismsIterator #O AllIsomorphismsNumber( , ) #O AllIsomorphisms( , ) ## DeclareOperation( "AllIsomorphismsIterator", [ IsGroup, IsGroup ] ); DeclareGlobalFunction( "DoAllIsomorphismsIterator" ); DeclareOperation( "AllIsomorphismsNumber", [ IsGroup, IsGroup ] ); DeclareOperation( "AllIsomorphisms", [ IsGroup, IsGroup ] ); ############################################################################# ## These functions provide an iterator for all subgroups of a group. ## #O AllSubgroupsIterator( ) #F DoAllSubgroupsIterator ## DeclareOperation( "AllSubgroupsIterator", [ IsGroup ] ); DeclareGlobalFunction( "DoAllSubgroupsIterator" ); ############################################################################# ## These functions provide ways of combining two iterators into one ## #O CartesianIterator( ) #F DoCartesianIterator #O UnorderedPairsIterator( ) #F DoUnorderedPairsIterator ## DeclareOperation( "CartesianIterator", [ IsIterator, IsIterator ] ); DeclareGlobalFunction( "DoCartesianIterator" ); DeclareOperation( "UnorderedPairsIterator", [ IsIterator ] ); DeclareGlobalFunction( "DoUnorderedPairsIterator" ); utils-0.86/lib/files.gi000644 000765 000024 00000007026 14621444241 020113 0ustar00christopherwensleystaff000000 000000 ############################################################################# ## #W files.gi GAP4 package `Utils' Sebastian Gutsche ## Max Horn ## Stefan Kohl #Y Copyright (C) 2015-2018, The GAP Group, ############################################################################# ## this function has been transferred from RCWA ## #F Log2HTML ( logfilename ) . . . . convert GAP logfile to XHTML 1.0 Strict ## InstallGlobalFunction( Log2HTML, function ( logfilename ) local outputname, s1, s2, header, footer, pos, lastlf, nextlf, crlf, prompt; if ARCH_IS_UNIX() then crlf := 1; else crlf := 2; fi; header := Concatenation( "\n\n", "\n\n\n\n ", logfilename, " \n \n", "\n\n\n\n
\n");
    footer := "
"; s1 := StringFile( logfilename ); pos := PositionSublist( s1, "gap>" ); prompt := "gap> "; s2 := ReplacedString( s1{[1..pos-1]}, "<", "<" ); while pos <> fail do s2 := Concatenation(s2,"",prompt,""); s2 := Concatenation(s2,""); nextlf := Position(s1,'\n',pos); prompt := "gap>"; if nextlf = fail then nextlf := Length(s1); fi; s2 := Concatenation(s2,ReplacedString(s1{[pos+5..nextlf-crlf]}, "<","<"),""); while nextlf < Length(s1) and s1[nextlf+1] = '>' do s2 := Concatenation(s2,"\n>", ""); lastlf := nextlf; nextlf := Position(s1,'\n',lastlf); if nextlf = fail then nextlf := Length(s1); fi; s2 := Concatenation(s2,ReplacedString(s1{[lastlf+2..nextlf-crlf]}, "<","<"),""); od; s2 := Concatenation( s2, "\n" ); pos := PositionSublist( s1, "\ngap>", nextlf-1 ); if pos = fail then pos := Length(s1); fi; if pos > nextlf then s2 := Concatenation(s2,"", ReplacedString( s1{[nextlf+1..pos-crlf]}, "<", "<"), "\n" ); fi; if pos > Length(s1) - 3 then break; fi; od; s2 := Concatenation( header, s2, footer ); logfilename := LowercaseString( logfilename ); if ( PositionSublist(logfilename,".log") <> fail ) then outputname := ReplacedString( logfilename, ".log", ".html" ); elif ( PositionSublist( logfilename, ".txt" ) <> fail ) then outputname := ReplacedString( logfilename, ".txt", ".html" ); else outputname := Concatenation( logfilename, ".html" ); fi; FileString( outputname, s2 ); end ); ############################################################################# ## #E files.gi . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/magma.gd000644 000765 000024 00000003213 14621444241 020060 0ustar00christopherwensleystaff000000 000000 ############################################################################## ## #W magma.gd GAP4 package `Utils' Max Horn ## Alexander Hulpke ## #Y Copyright (C) 2015-2017, The GAP Group ############################################################################# ## this function is new in Utils but calls functions which have been ## transferred from the main library or from private code ## #F ConvertToMagmaInputString( ) . . output a string readable by Magma ## DeclareGlobalFunction( "ConvertToMagmaInputString" ); ############################################################################# ## this function was originally authored by Alexander Hulpke; ## it has been transferred from other.gd in the main library; ## and was originally named MagmaInputString ## #O PermGroupToMagmaFormat( ) ## DeclareOperation( "PermGroupToMagmaFormat", [ IsPermGroup ] ); ############################################################################# ## this function was private code of Max Horn ## #O PcGroupToMagmaFormat( ) ## DeclareOperation( "PcGroupToMagmaFormat", [ IsPcGroup ] ); ############################################################################# ## this function was private code of Frank Lübeck ## #O MatrixGroupToMagmaFormat( ) ## DeclareOperation( "MatrixGroupToMagmaFormat", [ IsMatrixGroup ] ); ############################################################################# ## #E magma.gd . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/groups.gi000644 000765 000024 00000035275 14621444241 020337 0ustar00christopherwensleystaff000000 000000 ############################################################################# ## #W groups.gi GAP4 package `Utils' Stefan Kohl ## #Y Copyright (C) 2015-2022, The GAP Group ############################################################################# ## this function has been transferred from ResClasses ## #M Comm( [ , ] ) . . . for arguments enclosed in list brackets ## InstallOtherMethod( Comm, "for arguments enclosed in list brackets (ResClasses)", true, [ IsList ], 0, LeftNormedComm ); ############################################################################# ## this function has been transferred from ResClasses ## #M IsCommuting( , ) . . . . . . . . . . . . . . . . . fallback method ## InstallMethod( IsCommuting, "fallback method (ResClasses)", IsIdenticalObj, [ IsMultiplicativeElement, IsMultiplicativeElement ], 0, function ( a, b ) return a*b = b*a; end ); ############################################################################# ## this function has been transferred from ResClasses ## #M UpperFittingSeries( ) . . . . . . . . . . . . . . . . default method ## InstallMethod( UpperFittingSeries, "default method", true, [ IsGroup ], 0, function ( G ) local series, F, phi; if IsTrivial(FittingSubgroup(G)) then return [ TrivialSubgroup(G) ]; fi; F := FittingSubgroup(G); series := [ TrivialSubgroup(G), F ]; while F <> G do phi := NaturalHomomorphismByNormalSubgroup(G,F); F := PreImage(phi,FittingSubgroup(Image(phi))); if series[Length(series)] = F then break; fi; Add(series,F); od; return series; end ); ############################################################################# ## this function has been transferred from ResClasses ## #M LowerFittingSeries( ) . . . . . . . . . . . . . . . . default method ## InstallMethod( LowerFittingSeries, "default method", true, [ IsGroup ], 0, function ( G ) local series, F; series := [ G ]; F := G; while not IsTrivial(F) do F := Reversed(LowerCentralSeries(F))[1]; if series[Length(series)] = F then break; fi; Add(series,F); od; return series; end ); ############################################################################# ## this function has been transferred from ResClasses ## #M FittingLength( ) . . . . . . . . . . . . . . . . . . . default method ## InstallMethod( FittingLength, "default method", true, [ IsGroup ], 0, function ( G ) if not IsSolvableGroup(G) then return infinity; fi; if HasUpperFittingSeries(G) then return Length(UpperFittingSeries(G)) - 1; else return Length(LowerFittingSeries(G)) - 1; fi; end ); ############################################################################# ## this function has been transferred from RCWA ## #F ListOfPowers( , ) . . . . . . list of powers ^1 .. ^ ## InstallGlobalFunction( ListOfPowers, function ( g, exp ) local powers, n; powers := [g]; for n in [2..exp] do Add(powers,powers[n-1]*g); od; return powers; end ); ############################################################################# ## this function has been transferred from RCWA ## #M GeneratorsAndInverses( ) list of generators of and their inverses #M GeneratorsAndInverses( ) . . . . . . . . . . . . . . . . . for groups ## InstallMethod( GeneratorsAndInverses, "for groups", true, [ IsGroup ], 0, G -> Concatenation( GeneratorsOfGroup(G), List( GeneratorsOfGroup(G), g->g^-1 ) ) ); ############################################################################# ## temporary methods for SubdirectProductWithEmbeddings ## InstallGlobalFunction(SubdirectProductWithEmbeddings,function(G,H,ghom,hhom) local iso; if ( Range( ghom, G ) <> Range( hhom, H ) ) or ( Image( ghom, G ) <> Image( hhom, H ) ) then Error("the image groups are not the same"); fi; if not ( IsPermGroup(G) and IsPermGroup(H) ) or ( IsPcGroup(G) and IsPcGroup(H) ) then Error("methods only supplied forperm groups or pc groups"); fi; return SubdirectProductWithEmbeddingsOp( G, H, ghom, hhom ); end); ############################################################################# ## #M SubdirectProductWithEmbeddingsOp( , , , ) ## InstallMethod( SubdirectProductWithEmbeddingsOp,"groups", true, [ IsGroup, IsGroup, IsGroupHomomorphism, IsGroupHomomorphism ], 0, function( G, H, ghom, hhom ) local gc,hc,S,info; # try to enforce a common representation if not (IsFinite(G) and IsFinite(H)) then TryNextMethod(); fi; if IsSolvableGroup(G) and IsSolvableGroup(H) and not (IsPcGroup(G) and IsPcGroup(H)) then # enforce pc groups gc:=IsomorphismPcGroup(G); hc:=IsomorphismPcGroup(H); elif not (IsPermGroup(G) and IsPermGroup(H)) then # enforce perm groups gc:=IsomorphismPermGroup(G); hc:=IsomorphismPermGroup(H); else TryNextMethod(); fi; ghom := InverseGeneralMapping(gc)*ghom; hhom := InverseGeneralMapping(hc)*hhom; # the ...Op is installed for `IsGroupHomomorphism'. So we have to enforce # the filter to be set. if not IsGroupHomomorphism(ghom) or not IsGroupHomomorphism(hhom) then Error("mappings are not homomorphisms"); fi; S:=SubdirectProductWithEmbeddingsOp(Image(gc,G),Image(hc,H),ghom,hhom); info:=rec(groups:=[G,H], homomorphisms:=[ghom,hhom], projections:=[Projection(S,1)*InverseGeneralMapping(gc), Projection(S,2)*InverseGeneralMapping(hc)]); S:=Group(GeneratorsOfGroup(S)); SetSubdirectProductWithEmbeddingsInfo(S,info); return S; end); ############################################################################# ## #M Projection( , ) . . . . . . . . . . . . . . . . . make projection ## InstallMethod( Projection,"pc subdirect product", true, [ IsGroup and HasSubdirectProductWithEmbeddingsInfo, IsPosInt ], 0, function( S, i ) local prj, info; if not i in [1,2] then Error("only 2 embeddings"); fi; info := SubdirectProductWithEmbeddingsInfo( S ); if not IsBound(info.projections[i]) then TryNextMethod(); fi; return info.projections[i]; end); ############################################################################# ## #M Size( ) . . . . . . . . . . . . . . . . . . . . of subdirect product ## InstallMethod( Size,"subdirect product", true, [ IsGroup and HasSubdirectProductWithEmbeddingsInfo ], 0, function( S ) local info; info := SubdirectProductWithEmbeddingsInfo( S ); return Size( info.groups[ 1 ] ) * Size( info.groups[ 2 ] ) / Size( ImagesSource( info.homomorphisms[ 1 ] ) ); end ); ############################################################################# ## methods copied from gprdperm.gi ## ############################################################################# ## #M SubdirectProductWithEmbeddingsOp( , , , ) ## InstallMethod( SubdirectProductWithEmbeddingsOp, "permgroup", true, [ IsPermGroup, IsPermGroup, IsGroupHomomorphism, IsGroupHomomorphism ], 0, function( G1, G2, phi1, phi2 ) local S, # subdirect product of and , result gens, # generators of D, # direct product of and emb1, emb2, # embeddings of and into info, Dinfo,# info records gen; # one generator of or kernel of # make the direct product and the embeddings D := DirectProduct( G1, G2 ); emb1 := Embedding( D, 1 ); emb2 := Embedding( D, 2 ); # the subdirect product is generated by $(g_1,x_{g_1})$ where $g_1$ loops # over the generators of $G_1$ and $x_{g_1} \in G_2$ is abitrary such # that $g_1^{phi_1} = x_{g_1}^{phi_2}$ and by $(1,k_2)$ where $k_2$ loops # over the generators of the kernel of $phi_2$. gens := []; for gen in GeneratorsOfGroup( G1 ) do Add( gens, gen^emb1 * PreImagesRepresentativeNC(phi2,gen^phi1)^emb2 ); od; for gen in GeneratorsOfGroup( KernelOfMultiplicativeGeneralMapping( phi2 ) ) do Add( gens, gen ^ emb2 ); od; # and make the subdirect product S := GroupByGenerators( gens ); SetParent( S, D ); Dinfo := DirectProductInfo( D ); info := rec( groups := [G1, G2], homomorphisms := [phi1, phi2], olds := Dinfo.olds, news := Dinfo.news, perms := Dinfo.perms, projections := [] ); SetSubdirectProductWithEmbeddingsInfo( S, info ); return S; end ); ############################################################################# ## #R IsProjectionSubdirectProductWithEmbeddingsPermGroup( ) . projection onto factor ## DeclareRepresentation( "IsProjectionSubdirectProductWithEmbeddingsPermGroup", IsAttributeStoringRep and IsComponentObjectRep and IsGroupHomomorphism and IsSurjective and IsSPGeneralMapping, [ "component" ] ); ############################################################################# ## #M Projection( , ) . . . . . . . . . . . . . . . . . make projection ## InstallMethod( Projection,"perm subdirect product",true, [ IsPermGroup and HasSubdirectProductWithEmbeddingsInfo, IsPosInt ], 0, function( S, i ) local prj, info; info := SubdirectProductWithEmbeddingsInfo( S ); if IsBound( info.projections[i] ) then return info.projections[i]; fi; prj := Objectify( NewType( GeneralMappingsFamily( PermutationsFamily, PermutationsFamily ), IsProjectionSubdirectProductWithEmbeddingsPermGroup ), rec( component := i ) ); SetSource( prj, S ); info.projections[i] := prj; SetSubdirectProductWithEmbeddingsInfo( S, info ); return prj; end ); ############################################################################# ## #M Range( ) . . . . . . . . . . . . . . . . . . . . . . of projection ## InstallMethod( Range,"perm subdirect product projection", true, [ IsProjectionSubdirectProductWithEmbeddingsPermGroup ], 0, prj -> SubdirectProductWithEmbeddingsInfo( Source( prj ) ).groups[ prj!.component ] ); ############################################################################# ## #M ImagesRepresentative( , ) . . . . . . . . . . . . of projection ## InstallMethod( ImagesRepresentative,"perm subdirect product projection", FamSourceEqFamElm, [ IsProjectionSubdirectProductWithEmbeddingsPermGroup, IsMultiplicativeElementWithInverse ], 0, function( prj, g ) local info; info := SubdirectProductWithEmbeddingsInfo( Source( prj ) ); return RestrictedPermNC( g, info.news[ prj!.component ] ) ^ ( info.perms[ prj!.component ] ^ -1 ); end ); ############################################################################# ## #M PreImagesRepresentativeNC( , ) . . . . . . . . . . of projection ## InstallMethod( PreImagesRepresentativeNC,"perm subdirect product projection", FamRangeEqFamElm, [ IsProjectionSubdirectProductWithEmbeddingsPermGroup, IsMultiplicativeElementWithInverse ], 0, function( prj, img ) local S, elm, # preimage of under , result info, # info record phi1, phi2; # homomorphisms of components S := Source( prj ); info := SubdirectProductWithEmbeddingsInfo( S ); # get the homomorphism phi1 := info.homomorphisms[1]; phi2 := info.homomorphisms[2]; # compute the preimage if 1 = prj!.component then elm := img ^ info.perms[1] * PreImagesRepresentativeNC(phi2,img^phi1) ^ info.perms[2]; else elm := img ^ info.perms[2] * PreImagesRepresentativeNC(phi1,img^phi2) ^ info.perms[1]; fi; # return the preimage return elm; end ); ############################################################################# ## #M KernelOfMultiplicativeGeneralMapping( ) . . . . . . . of projection ## InstallMethod( KernelOfMultiplicativeGeneralMapping, "perm subdirect product projection",true, [ IsProjectionSubdirectProductWithEmbeddingsPermGroup ], 0, function( prj ) local D, i, info; D := Source( prj ); info := SubdirectProductWithEmbeddingsInfo( D ); i := 3 - prj!.component; return SubgroupNC( D, OnTuples ( GeneratorsOfGroup( KernelOfMultiplicativeGeneralMapping( info.homomorphisms[ i ] ) ), info.perms[ i ] ) ); end ); ############################################################################# ## #M ViewObj( ) . . . . . . . . . . . . . . . . . . . . view projection ## InstallMethod( ViewObj, "for projection from subdirect product", true, [ IsProjectionSubdirectProductWithEmbeddingsPermGroup ], 0, function( prj ) Print( Ordinal( prj!.component ), " projection of " ); View( Source( prj ) ); end ); ############################################################################# ## #M PrintObj( ) . . . . . . . . . . . . . . . . . . . print projection ## InstallMethod( PrintObj, "for projection from subdirect product", true, [ IsProjectionSubdirectProductWithEmbeddingsPermGroup ], 0, function( prj ) Print( "Projection( ", Source( prj ), ", ", prj!.component, " )" ); end ); ############################################################################# ## method copied from grpdpc.gi ## ############################################################################# ## #M SubdirectProductWithEmbeddingsOp( , , , ) ## InstallMethod( SubdirectProductWithEmbeddingsOp,"pcgroup", true, [ IsPcGroup, IsPcGroup, IsGroupHomomorphism, IsGroupHomomorphism ], 0, function( G, H, gh, hh ) local pg,ph,kg,kh,ig,ih,mg,mh,S,info; pg:=Pcgs(G); ph:=Pcgs(H); kg:=KernelOfMultiplicativeGeneralMapping(gh); kh:=KernelOfMultiplicativeGeneralMapping(hh); ig:=InducedPcgs(pg,kg); ih:=InducedPcgs(ph,kh); mg:=pg mod ig; mh:=List(mg,i->PreImagesRepresentativeNC(hh,Image(gh,i))); pg:=Concatenation(mg,ig,List(ih,i->One(G))); ph:=Concatenation(mh,List(ig,i->One(H)),ih); S:=SubdirProdPcGroups(G,pg,H,ph); pg:=GroupHomomorphismByImagesNC(S[1],G,S[2],pg); ph:=GroupHomomorphismByImagesNC(S[1],H,S[2],ph); S:=S[1]; info:=rec(groups:=[G,H], homomorphisms:=[gh,hh], projections:=[pg,ph]); SetSubdirectProductWithEmbeddingsInfo(S,info); return S; end); ############################################################################# ## #E groups.gi . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/string.gd000644 000765 000024 00000001371 14621444241 020307 0ustar00christopherwensleystaff000000 000000 ############################################################################## ## #W string.gd GAP4 package `Utils' Sebastian Gutsche ## Stefan Kohl ## #Y Copyright (C) 2015-2018, The GAP Group ############################################################################# ## this function has been transferred from ResClasses ## #F BlankFreeString( ) . . . . . . . . . . . . . string without blanks ## DeclareGlobalFunction( "BlankFreeString" ); ############################################################################# ## #E string.gd . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/download.gi000644 000765 000024 00000016664 14621444241 020630 0ustar00christopherwensleystaff000000 000000 ############################################################################## ## #W download.gi GAP4 package `Utils' Thomas Breuer ## #Y Copyright (C) 2022, The GAP Group ############################################################################# ## #V Download_Methods ## ## Use the following tools (in this order). ## ## - If the curlInterface package is available then call its ## 'DownloadURL' function. ## - If the URL starts with 'http://' and if the IO package is available ## then call the 'SingleHTTPRequest' function from this package. ## - If a 'wget' executable is available then call it. ## - If a 'curl' executable is available then call it. ## ## Note that currently the methods are *NOT* consistent in the case of ## failures: ## ## - The function 'SingleHTTPRequest' does not follow redirects as indicated ## by HTTP status codes 301 and 302. ## This happens for example if one asks for the file at ## 'http://www.gap-system.org/Packages/utils.html'. ## BindGlobal( "Download_Methods", [] ); Add( Download_Methods, rec( name:= "via DownloadURL (from the curlInterface package)", isAvailable:= {} -> IsPackageLoaded( "curlInterface" ) and CompareVersionNumbers( InstalledPackageVersion( "curlInterface" ), "2.3.0" ), download:= function( url, opt ) local res; opt:= ShallowCopy( opt ); if not IsBound( opt.failOnError ) then opt.failOnError:= true; fi; res:= ValueGlobal( "DownloadURL" )( url, opt ); if res.success = true and IsBound( opt.target ) and IsString( opt.target ) then FileString( opt.target, res.result ); Unbind( res.result ); fi; return res; end ) ); Add( Download_Methods, rec( name:= "via SingleHTTPRequest (from the IO package)", isAvailable:= {} -> IsBoundGlobal( "SingleHTTPRequest" ), download:= function( url, opt ) local rurl, pos, domain, uri, res; if not StartsWith( url, "http://" ) then return rec( success:= false, error:= "protocol is not http" ); fi; rurl:= ReplacedString( url, "http://", "" ); pos:= Position( rurl, '/' ); domain:= rurl{ [ 1 .. pos-1 ] }; uri:= rurl{ [ pos .. Length( rurl ) ] }; if IsBound( opt.target ) and IsString( opt.target ) then res:= ValueGlobal( "SingleHTTPRequest" )( domain, 80, "GET", uri, rec(), false, opt.target ); else res:= ValueGlobal( "SingleHTTPRequest" )( domain, 80, "GET", uri, rec(), false, false ); fi; if res.statuscode = 0 then return rec( success:= false, error:= res.status ); elif res.statuscode >= 400 then return rec( success:= false, error:= Concatenation( "HTTP error code ", res.statuscode ) ); elif not ( IsBound( opt.target ) and IsString( opt.target ) ) then return rec( success:= true, result:= res.body ); else return rec( success:= true ); fi; end ) ); Add( Download_Methods, rec( name:= "via wget", isAvailable:= function() local exec; exec:= Filename( DirectoriesSystemPrograms(), "wget" ); return exec <> fail and IsExecutableFile( exec ); end, download:= function( url, opt ) local res, outstream, exec, args, code; res:= ""; outstream:= OutputTextString( res, true ); exec:= Filename( DirectoriesSystemPrograms(), "wget" ); if IsBound( opt.target ) and IsString( opt.target ) then args:= [ "--quiet", "-O", opt.target, url ]; else args:= [ "--quiet", "-O", "-", url ]; fi; if IsBound( opt.verifyCert ) and opt.verifyCert = false then Add( args, "--no-check-certificate" ); fi; code:= Process( DirectoryCurrent(), exec, InputTextNone(), outstream, args ); CloseStream( outstream ); if code <> 0 then # wget may have created the target file; try to remove it if IsBound( opt.target ) and IsString( opt.target ) and IsExistingFile( opt.target ) and RemoveFile( opt.target ) <> true then Error( "Download cannot remove unwanted file ", opt.target ); fi; return rec( success:= false, error:= Concatenation( "Process returned ", String( code ) ) ); elif not ( IsBound( opt.target ) and IsString( opt.target ) ) then return rec( success:= true, result:= res ); else return rec( success:= true ); fi; end ) ); Add( Download_Methods, rec( name:= "via curl", isAvailable:= function() local exec; exec:= Filename( DirectoriesSystemPrograms(), "curl" ); return exec <> fail and IsExecutableFile( exec ); end, download:= function( url, opt ) local res, outstream, exec, args, code; res:= ""; outstream:= OutputTextString( res, true ); exec:= Filename( DirectoriesSystemPrograms(), "curl" ); args:= [ "--silent", "-L", "--fail" ]; if IsBound( opt.verifyCert ) and opt.verifyCert = false then Add( args, "-k" ); fi; Add( args, "--output" ); if IsBound( opt.target ) and IsString( opt.target ) then Add( args, opt.target ); else Add( args, "-" ); fi; Add( args, url ); code:= Process( DirectoryCurrent(), exec, InputTextNone(), outstream, args ); CloseStream( outstream ); if code <> 0 then return rec( success:= false, error:= Concatenation( "Process returned ", String( code ) ) ); elif not ( IsBound( opt.target ) and IsString( opt.target ) ) then return rec( success:= true, result:= res ); else return rec( success:= true ); fi; end ) ); ############################################################################# ## #M Download( [, ] ) ## ## Try to download the file described by the string , ## and return a record with the components 'success' ('true' or 'false'), ## and 'result' (a string, only if 'success' is 'true'), ## and 'error' (a string, only if 'success' is 'false'). ## InstallMethod( Download, [ "IsString" ], url -> Download( url, rec() ) ); InstallMethod( Download, [ "IsString", "IsRecord" ], function( url, opt ) local errors, r, res; # Set the default for 'verifyCert' if necessary. if not IsBound( opt.verifyCert ) and UserPreference( "utils", "DownloadVerifyCertificate" ) = false then opt.verifyCert:= false; fi; # Run over the methods. errors:= []; for r in Download_Methods do if r.isAvailable() then Info( InfoUtils, 2, "try Download method ", r.name ); res:= r.download( url, opt ); if res.success = true then return res; fi; Info( InfoUtils, 2, "Download method ", r.name, " failed with\n", "#I ", res.error ); Add( errors, Concatenation( r.name, ": ", res.error ) ); else Info( InfoUtils, 2, "Download method ", r.name, " is not available" ); fi; od; # No method was successful. if Length( errors ) = 0 then # No method was available, inform the user # that it is recommended to load or install some download tool. Info( InfoWarning, 1, "No 'Download' method is available.\n", "#I Please consider to install one of the tools, ", "see '?Download'" ); return rec( success:= false, error:= "no method was available" ); else # At least one method was tried but without success. return rec( success:= false, error:= JoinStringsWithSeparator( errors, "; " ) ); fi; end ); utils-0.86/lib/number.gi000644 000765 000024 00000015160 14621444241 020277 0ustar00christopherwensleystaff000000 000000 ############################################################################# ## #W number.gi GAP4 package `Utils' Stefan Kohl ## #Y Copyright (C) 2015-2018, The GAP Group ############################################################################# ## this function has been transferred from RCWA ## #F AllSmoothIntegers( , ) #F AllSmoothIntegers( , ) ## InstallGlobalFunction( AllSmoothIntegers, function ( maxp, maxn ) local extend, nums, primes, p; extend := function ( n, mini ) local i; if n > maxn then return; fi; Add(nums,n); for i in [mini..Length(primes)] do extend(primes[i]*n,i); od; end; if IsInt(maxp) then primes := Filtered([2..maxp],IsPrimeInt); elif IsList(maxp) and ForAll(maxp,p->IsInt(p) and IsPrimeInt(p)) then primes := maxp; else return fail; fi; if not IsPosInt(maxn) then return fail; fi; nums := []; extend(1,1); return Set(nums); end ); ############################################################################# ## this function has been transferred from RCWA ## #F NextProbablyPrimeInt( ) . . next integer passing `IsProbablyPrimeInt' ## InstallGlobalFunction( NextProbablyPrimeInt, function ( n ) if -3 = n then n := -2; elif -3 < n and n < 2 then n := 2; elif n mod 2 = 0 then n := n+1; else n := n+2; fi; while not IsProbablyPrimeInt(n) do if n mod 6 = 1 then n := n+4; else n := n+2; fi; od; return n; end ); ############################################################################# ## this function has been transferred from RCWA ## #F RestrictedPartitionsWithoutRepetitions( , ) ## ## Given a positive integer n and a set of positive integers S, this func- ## tion returns a list of all partitions of n into distinct elements of S. ## The only difference to `RestrictedPartitions' is that no repetitions are ## allowed. ## InstallGlobalFunction( RestrictedPartitionsWithoutRepetitions, function ( n, S ) local look, comps; look := function ( comp, remaining_n, remaining_S ) local newcomp, newremaining_n, newremaining_S, part, l; l := Reversed(remaining_S); for part in l do newcomp := Concatenation(comp,[part]); newremaining_n := remaining_n - part; if newremaining_n = 0 then Add(comps,newcomp); else newremaining_S := Set(Filtered(remaining_S, s->s [] then look(newcomp,newremaining_n,newremaining_S); fi; fi; od; end; comps := []; look([],n,S); return comps; end ); ############################################################################# ## this function has been transferred from RCWA ## #F PrimeNumbersIterator( ) #F PrimeNumbersIterator( chunksize ) ## InstallGlobalFunction( PrimeNumbersIterator, function ( arg ) local next, copy, chunksize, maxdiv, nrdivs, offset; if Length(arg) >= 1 and IsPosInt(arg[1]) then chunksize := Maximum(arg[1],100); # must be bigger than largest else chunksize := 10000000; fi; # prime gap in range looped over maxdiv := RootInt(chunksize); offset := List(Filtered([2..maxdiv],IsPrimeInt),p->[p,0]); nrdivs := Length(offset); return IteratorByFunctions( rec( NextIterator := function ( iter ) local sieve, range, p, q, pos, endpos, maxdiv_old, maxdiv, i, j; if iter!.index = 0 then sieve := ListWithIdenticalEntries(iter!.chunksize,0); if iter!.n = 0 then sieve[1] := 1; fi; for i in [1..iter!.nrdivs] do p := iter!.offset[i][1]; if p > iter!.n then pos := 2 * p; else pos := iter!.offset[i][2]; fi; if pos = 0 then pos := p; fi; endpos := pos + Int((iter!.chunksize-pos)/p) * p; if pos <= iter!.chunksize then range := [pos,pos+p..endpos]; if IsRangeRep(range) then ADD_TO_LIST_ENTRIES_PLIST_RANGE(sieve,range,1); else for j in range do sieve[j] := sieve[j] + 1; od; fi; fi; if endpos <= iter!.chunksize then iter!.offset[i][2] := endpos + p - iter!.chunksize; else iter!.offset[i][2] := iter!.offset[i][2] - iter!.chunksize; fi; od; iter!.primepos := Positions(sieve,0); fi; iter!.index := iter!.index + 1; p := iter!.n + iter!.primepos[iter!.index]; iter!.p := p; iter!.pi := iter!.pi + 1; if iter!.index = Length(iter!.primepos) then iter!.index := 0; iter!.n := iter!.n + iter!.chunksize; maxdiv_old := iter!.maxdiv; iter!.maxdiv := RootInt(iter!.n + iter!.chunksize); for q in Filtered([maxdiv_old+1..iter!.maxdiv],IsPrimeInt) do Add(iter!.offset,[q,(q-iter!.n) mod q]); od; iter!.nrdivs := Length(iter!.offset); fi; return p; end, ShallowCopy := function ( iter ) return rec( chunksize := iter!.chunksize, n := iter!.n, p := iter!.p, pi := iter!.pi, index := iter!.index, primepos := ShallowCopy(iter!.primepos), nrdivs := iter!.nrdivs, maxdiv := iter!.maxdiv, offset := StructuralCopy(iter!.offset) ); end, IsDoneIterator := ReturnFalse, chunksize := chunksize, n := 0, p := 0, pi := 0, index := 0, primepos := [], nrdivs := nrdivs, maxdiv := maxdiv, offset := offset ) ); end ); ############################################################################# ## this function has been transferred from RCWA ## #M AllProducts( , ) . . all products of -tuples of elements of #M AllProducts( , ) . . . . . . . . . . . . . . . . . . . . for lists ## InstallMethod( AllProducts, "for lists (RCWA)", ReturnTrue, [ IsList, IsPosInt ], 0, function(l,k) return List(Tuples(l,k),Product); end ); ############################################################################# ## #E number.gi . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/latex.gd000644 000765 000024 00000001660 14621444241 020117 0ustar00christopherwensleystaff000000 000000 ############################################################################## ## #W latex.gd GAP4 package `Utils' Stefan Kohl ## #Y Copyright (C) 2015-2016, The GAP Group ############################################################################# ## this function has been transferred from ResClasses ## #F IntOrInfinityToLaTeX( n ) . LaTeX string representing integer or infinity ## DeclareGlobalFunction( "IntOrInfinityToLaTeX" ); ############################################################################# ## this function has been transferred from RCWA ## #F LaTeXStringFactorsInt( ) ## ## Returns the prime factorization of the integer as a string in LaTeX ## format. ## DeclareGlobalFunction( "LaTeXStringFactorsInt" ); ############################################################################# ## #E latex.gd . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/lists.gd000644 000765 000024 00000003737 14621444241 020147 0ustar00christopherwensleystaff000000 000000 ############################################################################## ## #W lists.gd GAP4 package `Utils' Stefan Kohl ## #Y Copyright (C) 2015-2018, The GAP Group ############################################################################# ## these functions have been transferred from ResClasses ## #F DifferencesList( ) . . . . differences of consecutive list entries #F QuotientsList( ) . . . . . . quotients of consecutive list entries #F FloatQuotientsList( ) . . . . . . . . . . . . dito, but as floats ## DeclareGlobalFunction( "DifferencesList" ); DeclareGlobalFunction( "QuotientsList" ); DeclareGlobalFunction( "FloatQuotientsList" ); ############################################################################# ## this function has been transferred from ResClasses ## #F RandomCombination( S, k ) ## ## Returns a random unordered -tuple of distinct elements of the set . ## DeclareOperation( "RandomCombination", [ IsListOrCollection, IsPosInt ] ); ############################################################################# ## this function has been transferred from RCWA ## #F SearchCycle( ) . . . a utility function for detecting cycles in lists ## DeclareGlobalFunction( "SearchCycle" ); ############################################################################# ## these functions have been have been transferred from XMod ## #O DistinctRepresentatives( ) #O CommonRepresentatives( , ) #O CommonTransversal( , ) #O IsCommonTransversal( , , ) ## DeclareOperation( "DistinctRepresentatives", [ IsList ] ); DeclareOperation( "CommonRepresentatives", [ IsList, IsList ] ); DeclareOperation( "CommonTransversal", [ IsGroup, IsGroup ] ); DeclareOperation( "IsCommonTransversal", [ IsGroup, IsGroup, IsList ] ); ############################################################################# ## #E lists.gd . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/lcset.gd000644 000765 000024 00000002646 14621444241 020121 0ustar00christopherwensleystaff000000 000000 ############################################################################## ## #W lcset.gd GAP4 package `Utils' Chris Wensley ## #Y Copyright (C) 2015-2022, The GAP Group ############################################################################# ## #C IsLeftCoset( ) . . . . . . . . . . . . . . coset of the form gU #R IsLeftCosetDefaultRep #V IsLeftCosetFamily #T IsLeftCosetType ## DeclareCategory( "IsLeftCoset", IsDomain and IsExternalOrbit ); DeclareRepresentation( "IsLeftCosetDefaultRep", IsComponentObjectRep and IsAttributeStoringRep and IsLeftCoset, [ "Representative", "ActingDomain" ] ); BindGlobal( "IsLeftCosetFamily", NewFamily( "IsLeftCosetFamily", IsMultiplicativeElementWithInverse ) ); BindGlobal( "IsLeftCosetType", NewType( IsLeftCosetFamily, IsLeftCosetDefaultRep ) ); ############################################################################# ## #O LeftCoset( , ) . . . . . . . element g acts on the left of group U ## DeclareOperation( "LeftCoset", [ IsObject, IsGroup ] ); ############################################################################# ## #O LeftCosets( , ) . . . . . . . . . . . . . . . cosets gU for g in G #O LeftCosetsNC( , ) . . . . . . . . . . . . . . cosets gU for g in G ## DeclareGlobalFunction( "LeftCosets" ); DeclareOperation( "LeftCosetsNC", [ IsGroup, IsGroup ] ); utils-0.86/lib/print.gi000644 000765 000024 00000011204 14621444241 020136 0ustar00christopherwensleystaff000000 000000 ############################################################################# ## #W print.gi GAP4 package `Utils' Chris Wensley ## #Y Copyright (C) 2015-2024, The GAP Group ############################################################################# ## these operations have not been transferred from any other package ## #M PrintSelection( ) #M PrintSelectionFromList( , , , ) #M PrintSelectionFromIterator( , , , ) ## InstallGlobalFunction( "PrintSelection", function( arg ) local nargs, obj, first, step, last, L, ok; nargs := Length( arg ); obj := arg[1]; if ( ( nargs = 1 ) or ( nargs > 4 ) ) then Error( "expecting 2, 3 or 4 arguments" ); elif ( nargs = 2 ) then L := arg[2]; if IsList( L ) then ok := ForAll( L, IsPosInt ); if not ok then Error( "L must only contain positive integers" ); fi; if IsList( obj ) then PrintSelectionFromListByList( obj, L ); elif IsIterator( obj ) then PrintSelectionFromIteratorByList( obj, L ); else PrintSelectionFromIteratorByList( Iterator( obj ), L ); fi; else Error( "second argument not a list" ); fi; else first := arg[2]; step := arg[3]; if ( nargs = 3 ) then if IsList( obj ) then last := Length( obj ); else last := infinity; fi; else last := arg[4]; fi; if IsList( obj ) then PrintSelectionFromList( obj, first, step, last ); elif IsIterator( obj ) then PrintSelectionFromIterator( obj, first, step, last ); else PrintSelectionFromIterator( Iterator( obj ), first, step, last ); fi; fi; end ); InstallMethod( PrintSelectionFromList, "generic method for lists", true, [ IsList, IsPosInt, IsPosInt, IsPosInt ], 0, function( L, first, step, last ) local len, i; len := Minimum( Length( L ), last ); if ( first <= len ) then Print( first, " : ", L[first], "\n" ); len := len-first; i := first; while ( len >= step ) do i := i+step; Print( i, " : ", L[i], "\n" ); len := len-step; od; fi; end ); InstallOtherMethod( PrintSelectionFromIterator, "generic method for iterators", true, [ IsIterator, IsPosInt, IsPosInt, IsAdditiveElement ], 0, function( iter0, first, step, last ) local iter, pos, i, x; if ( last < first ) then Error( "expecting first <= last" ); fi; iter := ShallowCopy( iter0 ); i := 0; while( ( i < first ) and not IsDoneIterator( iter ) ) do i := i+1; x := NextIterator( iter ); od; if ( i = first ) then pos := first; Print( pos, " : ", x, "\n" ); fi; while ( not IsDoneIterator( iter ) and ( pos < last ) ) do while ( not IsDoneIterator( iter ) and ( pos < last ) ) do i := 0; while( ( i ) ## ## This auxiliary function assigns the record components of ## to global variables with the same names. ## InstallGlobalFunction( AssignGlobals, function ( record ) local names, name; names := RecNames(record); for name in names do if IsBoundGlobal(name) then if IsReadOnlyGlobal(name) then MakeReadWriteGlobal(name); Info(InfoWarning,1,"The read-only global variable ",name, " has been overwritten."); else Info(InfoUtils,1,"The global variable ",name, " has been overwritten."); fi; UnbindGlobal(name); fi; BindGlobal(name,record.(name)); MakeReadWriteGlobal(name); od; Print("The following global variables have been assigned:\n", Set(names),"\n"); end ); InstallGlobalFunction( OptionRecordWithDefaults, function(default, useroptions) local name, ret; ret := rec(); if IsList(useroptions) then if IsEmpty(useroptions) then return default; elif Length(useroptions) = 1 then useroptions := useroptions[1]; else ErrorNoReturn("Too many arguments for function"); fi; fi; if not IsRecord(useroptions) then ErrorNoReturn("Options should be a record"); fi; ret := ShallowCopy(default); for name in RecNames(useroptions) do if not IsBound(default.(name)) then ErrorNoReturn(Concatenation("Unknown option: " , name)); else ret.(name) := useroptions.(name); fi; od; return ret; end); ############################################################################# ## #E record.gi . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/matrix.gd000644 000765 000024 00000001023 14621444241 020277 0ustar00christopherwensleystaff000000 000000 ############################################################################ ## #W matrix.gd GAP4 package `Utils' Chris Wensley ## #Y Copyright (C) 2023, The GAP Group ############################################################################# ## This function ... ## #O DirectSumDecompositionMatrices( ) ## DeclareOperation( "DirectSumDecompositionMatrices", [ IsMatrixOrMatrixObj ] ); DeclareOperation( "DirectSumBlock", [ IsMatrixOrMatrixObj, IsInt, IsList, IsList ] ); utils-0.86/lib/files.gd000644 000765 000024 00000002101 14621444241 020073 0ustar00christopherwensleystaff000000 000000 ############################################################################## ## #W files.gd GAP4 package `Utils' Sebastian Gutsche ## Max Horn ## Stefan Kohl #Y Copyright (C) 2015-2018, The GAP Group ############################################################################# ## this function has been transferred from RCWA ## #F Log2HTML ( logfilename ) ## ## Utility to convert GAP log files to XHTML 1.0 Strict. ## ## Usage: ## ## - Issue Log2HTML( ). The extension of the input file must ## be *.log. The name of the output file is the same as the one of the ## input file except that the extension *.log is replaced by *.html. ## ## - Adapt the style file rcwa/doc/gaplog.css to your taste. ## DeclareGlobalFunction( "Log2HTML" ); ############################################################################# ## #E files.gd . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/iterator.gi000644 000765 000024 00000020715 14621444241 020642 0ustar00christopherwensleystaff000000 000000 ############################################################################# ## #W iterator.gi GAP4 package `Utils' Chris Wensley ## #Y Copyright (C) 2015-2019, The GAP Group ############################################################################## ## #M AllIsomorphismsIterator . . . . . . . . . . . . . . . for a pair of groups #M DoAllIsomorphismsIterator . . . . . . . . . . . . . . for a pair of groups #M AllIsomorphismsNumber . . . . . . . . . . . . . . . . for a pair of groups #M AllIsomorphisms . . . . . . . . . . . . . . . . . . . for a pair of groups ## BindGlobal( "NextIterator_AllIsomorphisms", function ( iter ) local a; if not IsDoneIterator( iter ) then a := NextIterator( iter!.autoIterator ); return CompositionMapping( iter!.firstiso, a ); fi; Error( "iterator is exhausted" ); end ); BindGlobal( "IsDoneIterator_AllIsomorphisms", iter -> IsDoneIterator( iter!.autoIterator ) ); BindGlobal( "ShallowCopy_AllIsomorphisms", iter -> rec( firstiso := iter!.firstiso, autoIterator := ShallowCopy( iter!.autoIterator ) ) ); InstallGlobalFunction( "DoAllIsomorphismsIterator", function( G, H ) local iso, autoiter, iter; if not IsGroup( G ) and IsGroup( H ) then Error( "G,H should be groups" ); fi; iso := IsomorphismGroups( G, H ); if ( iso = fail ) then ## there are no isomorphisms so return an empty iterator return IteratorList( [ ] ); fi; autoiter := Iterator( AutomorphismGroup( G ) ); iter := IteratorByFunctions( rec( firstiso := iso, autoIterator := ShallowCopy( autoiter ), NextIterator := NextIterator_AllIsomorphisms, IsDoneIterator := IsDoneIterator_AllIsomorphisms, ShallowCopy := ShallowCopy_AllIsomorphisms ) ); return iter; end ); InstallMethod( AllIsomorphismsIterator, "for a pair of groups", [ IsGroup, IsGroup ], 0, function( G, H ) return DoAllIsomorphismsIterator( G, H ); end ); InstallMethod( AllIsomorphisms, "for a pair of groups", [ IsGroup, IsGroup ], 0, function( G, H ) local iter, L, iso; iter := AllIsomorphismsIterator( G, H ); if IsDoneIterator( iter ) then return [ ]; fi; L := [ ]; for iso in iter do Add( L, iso ); od; return L; end ); InstallMethod( AllIsomorphismsNumber, "for a pair of groups", [ IsGroup, IsGroup ], 0, function( G, H ) local iter, n, iso; iter := AllIsomorphismsIterator( G, H ); if IsDoneIterator( iter ) then return 0; fi; n := 0; for iso in iter do n := n+1; od; return n; end ); ############################################################################## ## #M AllSubgroupsIterator . . . . . . . . . . . . . . . . . . . . . for a group #M DoAllSubgroupsIterator . . . . . . . . . . . . . . . . . . . . for a group ## BindGlobal( "NextIterator_AllSubgroups", function ( iter ) local class, i; if IsDoneIterator( iter!.subgpIterator ) then class := NextIterator( iter!.classIterator ); iter!.class := class; iter!.subgpIterator := IteratorList( [1..Size(class)] ); fi; i := NextIterator( iter!.subgpIterator ); return iter!.class[i]; end ); BindGlobal( "IsDoneIterator_AllSubgroups", iter -> IsDoneIterator( iter!.classIterator ) and IsDoneIterator( iter!.subgpIterator ) ); BindGlobal( "ShallowCopy_AllSubgroups", iter -> rec( class := iter!.class, classIterator := ShallowCopy( iter!.classIterator ), subgpIterator := ShallowCopy( iter!.subgpIterator ) ) ); InstallGlobalFunction( "DoAllSubgroupsIterator", function( G ) local lat, ccs, classIterator, subgpIterator, iter; lat := LatticeSubgroups( G ); ccs := ConjugacyClassesSubgroups( lat ); classIterator := IteratorList( ccs ); subgpIterator := 0; iter := IteratorByFunctions( rec( class := 0, classIterator := ShallowCopy( classIterator ), subgpIterator := IteratorList( [ ] ), NextIterator := NextIterator_AllSubgroups, IsDoneIterator := IsDoneIterator_AllSubgroups, ShallowCopy := ShallowCopy_AllSubgroups ) ); return iter; end ); InstallMethod( AllSubgroupsIterator, "for a group", [ IsGroup ], 0, function( G ) return DoAllSubgroupsIterator( G ); end ); ############################################################################## ## #M CartesianIterator . . . . . . . . . . . . . . . . . . . for two iterators #M DoCartesianIterator . . . . . . . . . . . . . . . . . . for two iterators ## BindGlobal( "NextIterator_CartesianIterator", function ( iter ) local first, second; if IsDoneIterator( iter!.Iterator2Copy ) then iter!.Iterator2Copy := ShallowCopy( iter!.Iterator2 ); first := NextIterator( iter!.Iterator1 ); iter!.first := first; else first := iter!.first; fi; second := NextIterator( iter!.Iterator2Copy ); return [ first, second ]; end ); BindGlobal( "IsDoneIterator_CartesianIterator", iter -> IsDoneIterator( iter!.Iterator1 ) and IsDoneIterator( iter!.Iterator2Copy ) ); BindGlobal( "ShallowCopy_CartesianIterator", iter -> rec( first := iter!.first, Iterator1 := ShallowCopy( iter!.Iterator1 ), Iterator2 := ShallowCopy( iter!.Iterator2 ), Iterator2Copy := ShallowCopy( iter!.Iterator2Copy ) ) ); InstallGlobalFunction( "DoCartesianIterator", function( iter1, iter2 ) local iter; if IsDoneIterator( iter1 ) or IsDoneIterator( iter2 ) then return IteratorList( [ ] ); fi; iter := IteratorByFunctions( rec( first := 0, Iterator1 := ShallowCopy( iter1 ), Iterator2 := ShallowCopy( iter2 ), Iterator2Copy := IteratorList( [ ] ), NextIterator := NextIterator_CartesianIterator, IsDoneIterator := IsDoneIterator_CartesianIterator, ShallowCopy := ShallowCopy_CartesianIterator ) ); return iter; end ); InstallMethod( CartesianIterator, "for two iterators", [ IsIterator, IsIterator ], 0, function( iter1, iter2 ) return DoCartesianIterator( iter1, iter2 ); end ); ############################################################################## ## #M UnorderedPairsIterator . . . . . . . . . . . . . . . . . . for an iterator #M DoUnorderedPairsIterator . . . . . . . . . . . . . . . . . for an iterator ## BindGlobal( "NextIterator_UnorderedPairs", function ( iter ) local first, second; if ( iter!.pos1 = iter!.pos2 ) then iter!.Iterator1Copy := ShallowCopy( iter!.Iterator1 ); iter!.pos1 := 0; second := NextIterator( iter!.Iterator2 ); iter!.pos2 := iter!.pos2 + 1; iter!.second := second; else second := iter!.second; fi; first := NextIterator( iter!.Iterator1Copy ); iter!.pos1 := iter!.pos1 + 1; return [ first, second ]; end ); BindGlobal( "IsDoneIterator_UnorderedPairs", iter -> IsDoneIterator( iter!.Iterator2 ) and IsDoneIterator( iter!.Iterator1Copy ) ); BindGlobal( "ShallowCopy_UnorderedPairs", iter -> rec( pos1 := iter!.pos1, pos2 := iter!.pos2, second := iter!.second, Iterator1 := ShallowCopy( iter!.Iterator1 ), Iterator2 := ShallowCopy( iter!.Iterator2 ), Iterator1Copy := ShallowCopy( iter!.Iterator1Copy ) ) ); InstallGlobalFunction( "DoUnorderedPairsIterator", function( iter0 ) local iter; if IsDoneIterator( iter0 ) then return IteratorList( [ ] ); fi; iter := IteratorByFunctions( rec( pos1 := 0, pos2 := 0, second := 0, Iterator1 := ShallowCopy( iter0 ), Iterator2 := ShallowCopy( iter0 ), Iterator1Copy := ShallowCopy( iter0 ), NextIterator := NextIterator_UnorderedPairs, IsDoneIterator := IsDoneIterator_UnorderedPairs, ShallowCopy := ShallowCopy_UnorderedPairs ) ); return iter; end ); InstallMethod( UnorderedPairsIterator, "for an iterator", [ IsIterator ], 0, function( iter0 ) return DoUnorderedPairsIterator( iter0 ); end ); utils-0.86/lib/maps.gd000644 000765 000024 00000004371 14621444241 017744 0ustar00christopherwensleystaff000000 000000 ############################################################################## ## #W maps.gd GAP4 package `Utils' Stefan Kohl ## Chris Wensley #Y Copyright (C) 2015-2019, The GAP Group ############################################################################# ## these three functions were in RCWA and were initially transferred but, ## to simplify the situation, just EpimorphismByGenerators has been retained ## #O EpimorphismByGenerators( , ) . epi: gen's of ->gen's of #O EpimorphismByGeneratorsNC( , ) . . NC version as underlying oper #M EpimorphismByGeneratorsNC( , ) . . . . . . . . . . . . for groups ## DeclareOperation( "EpimorphismByGenerators", [ IsDomain, IsDomain ] ); ############################################################################# ## #O Pullback( , ) #A PullbackInfo( ) ## DeclareOperation( "Pullback", [ IsGroupHomomorphism, IsGroupHomomorphism ] ); DeclareAttribute( "PullbackInfo", IsGroup, "mutable" ); ############################################################################# ## #O CentralProduct( , , , ) #A CentralProductInfo( ) ## DeclareOperation( "CentralProduct", [ IsGroup, IsGroup, IsGroup, IsGroupHomomorphism ] ); DeclareAttribute( "CentralProductInfo", IsGroup, "mutable" ); ############################################################################# ## these functions compute idempotent endomorphisms on G with image R ## #O IdempotentEndomorphisms( ) #A IdempotentEndomorphismsWithImage( , ) #A IdempotentEndomorphismsData( ) ## DeclareOperation( "IdempotentEndomorphisms", [ IsGroup ] ); DeclareOperation( "IdempotentEndomorphismsWithImage", [ IsList, IsGroup ] ); DeclareAttribute( "IdempotentEndomorphismsData", IsGroup ); ############################################################################# ## #O DirectProductOfFunctions( , , , ) #O DirectProductOfAutomorphismGroups( , ) ## DeclareOperation( "DirectProductOfFunctions", [ IsGroup, IsGroup, IsGroupHomomorphism, IsGroupHomomorphism ] ); DeclareOperation( "DirectProductOfAutomorphismGroups", [ IsGroup, IsGroup ] ); utils-0.86/lib/magma.gi000644 000765 000024 00000020265 14621444241 020073 0ustar00christopherwensleystaff000000 000000 ############################################################################# ## #W magma.gi GAP4 package `Utils' Max Horn ## Alexander Hulpke ## Frank Lübeck ## #Y Copyright (C) 2015-2022, The GAP Group ############################################################################# ## #F ConvertToMagmaInputString( ) . . output a string readable by Magma ## InstallGlobalFunction( ConvertToMagmaInputString, function ( arg ) local len, obj, s, str; len := Length( arg ); obj := arg[1]; if not IsGroup( obj ) then Error( "input obj should be a group" ); elif IsPcGroup( obj ) then str := PcGroupToMagmaFormat( obj ); elif IsPermGroup( obj ) then str := PermGroupToMagmaFormat( obj ); elif IsMatrixGroup( obj ) then str := MatrixGroupToMagmaFormat( obj ); elif IsFpGroup( obj ) then Error( "no conversion function yet available for fp-groups" ); ## str := FpGroupToMagmaFormat( obj ); else Error( "no conversion function yet available for this group" ); fi; if ( len > 1 ) then s := arg[2]; if IsMatrixGroup( obj ) then str := ReplacedString( str, "sub", Concatenation(s," := sub") ); else str := Concatenation( s, " := ", str ); fi; fi; return str; end ); ############################################################################# ## this function was originally authored by Alexander Hulpke; ## has been transferred from other.gi in the main library; ## and was originally named MagmaInputString ## InstallMethod( PermGroupToMagmaFormat, "perm group", true, [IsPermGroup], 0, function( g ) local s, i, nf; ## s := ShallowCopy(s); ## Append(s,":=PermutationGroup<"); s := "PermutationGroup<"; Append( s, String( LargestMovedPoint(g) ) ); Add( s, '|' ); nf := false; for i in GeneratorsOfGroup(g) do if nf then Append( s, ",\n" ); fi; nf:=true; Append( s, String(i) ); od; Append( s, ">;\n" ); return s; end ); ############################################################################# ## this function was private code of Max Horn ## InstallMethod( PcGroupToMagmaFormat, "pc-group", true, [IsPcGroup], 0, function(G) local str, g, n, r, gens, rels, i, j; g := Pcgs(G); n := Length(g); r := RelativeOrders(g); gens := List(g,String); str := "PolycyclicGroup< "; Append(str, JoinStringsWithSeparator(gens)); Append(str, " |\n"); rels := []; for i in [1..n] do if r[i] = 0 then continue; fi; if IsOne(g[i]^r[i]) then Add(rels, Concatenation(String(g[i]), "^", String(r[i]))); else Add(rels, Concatenation(String(g[i]), "^", String(r[i]), " = ", String(g[i]^r[i]))); fi; od; for i in [2..n] do for j in [1..i-1] do if g[i]*g[j] <> g[j]*g[i] then Add(rels, Concatenation(String(g[i]), "^", String(g[j]), " = ", String(g[i]^g[j]))); if r[j] = 0 then Add(rels, Concatenation(String(g[i]), "^", String(g[j]), "^-1 = ", String(g[i]^(g[j]^-1)))); fi; fi; od; od; Append(str, JoinStringsWithSeparator(rels, ",\n")); Append(str, "\n>;\n"); return str; end ); ############################################################################### ## This function has been converted from private code of Frank Lübeck. ## His GapToMagma(file,mats) wrote Magma code to a file 'file' for the group ## generated by a list 'mats' of matrices over same finite field. ## In this revised form the function inputs a matrix group over a finite ## field and outputs a string which can be read by Magma. ## InstallMethod( MatrixGroupToMagmaFormat, "matrix group over a finite field", true, [ IsMatrixGroup ], 0, function(G) local mats, nr, nc, F, tab, p, q, e, w, wp, pow, zero, one, val, cpn, npl, NmrIter, FinalIter, i, j, cno, isint, matrix, str; mats := GeneratorsOfGroup( G ); F := Field( Flat( mats ) ); p := Characteristic( F ); e := DegreeOverPrimeField( F ); q := Size( F ); w := PrimitiveElement( F ); zero := Zero( F ); one := One( F ); matrix := mats[1]; nr := Length( matrix ); nc := Length( matrix[1] ); str := Concatenation( "F := ", String(F), ";\n" ); Append( str, Concatenation( "P := GL(", String(nr), ",F);\n" ) ); matrix := mats; if ( e > 1 ) then Append( str, "w := PrimitiveElement(F);\n" ); fi; pow := (q-1)/(p-1); wp := w^pow; tab := [ ]; for i in [1..p-1] do tab[ LogFFE( i*one, wp ) + 1 ] := i; od; if ( p < 10 ) then cpn := 3; elif ( p < 100 ) then cpn := 4; elif ( p < 1000 ) then cpn := 5; elif ( p < 10000 ) then cpn := 6; else cpn := 7; fi; if ( e > 1 ) then cpn := cpn + 2; fi; npl := Int(76/(cpn)); ## should be 78, but it overflowed. Append( str, "gens := [\n"); FinalIter := Length( mats ); NmrIter := 0; for matrix in mats do NmrIter := NmrIter + 1; nr := Length( matrix ); nc := Length( matrix[1] ); Append( str, "P![" ); for i in [1..nr] do cno := 0; for j in [1..nc] do if ( matrix[i][j] = zero ) then val := 0; isint := true; else val := LogFFE( matrix[i][j], w ); if ( val mod pow = 0 ) then val := tab[ LogFFE( matrix[i][j], wp ) + 1]; isint := true; else isint := false; fi; fi; if ( q<10 ) or ( q<100 and val>=10 ) or ( q<1000 and val>=100 ) or ( q<10000 and val>=1000 ) or ( val>=10000 ) then if not (i = 1 and j = 1) then Append( str, "," ); fi; elif ( q<100 ) or ( q<1000 and val>=10 ) or ( q<10000 and val>=100 ) or ( val>=1000 ) then if not (i = 1 and j = 1) then Append( str, " , " ); fi; elif ( q<1000 ) or ( q<10000 and val>=10 ) or ( val>=100 ) then if not (i = 1 and j = 1) then Append( str, " , " ); fi; elif ( q<10000 ) or ( val>=10 ) then if not (i = 1 and j = 1) then Append( str, " , " ); fi; else if not (i = 1 and j = 1) then Append( str, " , " ); fi; fi; if ( e > 1 ) then if isint then Append( str, " " ); else Append( str, "w^" ); fi; fi; Append( str, String(val) ); cno := cno+1; if ( cno>=npl and j FinalIter ) then Append( str, "],\n" ); else Append( str, "]\n" ); fi; od; Append( str, "];\n" ); Append( str, "sub

;\n" ); return str; end ); ############################################################################# ## #E magma.gi . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/groups.gd000644 000765 000024 00000005113 14621444241 020316 0ustar00christopherwensleystaff000000 000000 ############################################################################## ## #W groups.gd GAP4 package `Utils' Stefan Kohl ## #Y Copyright (C) 2015-2016, The GAP Group ############################################################################# ## this function has been transferred from ResClasses ## #O IsCommuting( , ) . checks whether two group elements etc. commute ## DeclareOperation( "IsCommuting", [ IsMultiplicativeElement, IsMultiplicativeElement ] ); ############################################################################# ## these functions have been transferred from ResClasses ## #A UpperFittingSeries( ) . . . . . . . . . . upper fitting series of #A LowerFittingSeries( ) . . . . . . . . . . lower fitting series of #A FittingLength( ) . . . . . . . . . . . . . . . fitting length of ## ## The upper and lower Fitting series and the Fitting length of a solvable ## group, as described here: https://en.wikipedia.org/wiki/Fitting_length ## if not IsBound( UpperFittingSeries ) then DeclareAttribute( "UpperFittingSeries", IsGroup ); fi; if not IsBound( LowerFittingSeries ) then DeclareAttribute( "LowerFittingSeries", IsGroup ); fi; if not IsBound( FittingLength ) then DeclareAttribute( "FittingLength", IsGroup ); fi; ############################################################################# ## this function has been transferred from RCWA ## #F ListOfPowers( , ) . . . . . . list of powers ^1 .. ^ ## DeclareGlobalFunction( "ListOfPowers" ); ############################################################################# ## this function has been transferred from RCWA ## #O GeneratorsAndInverses( ) list of generators of and their inverses #M GeneratorsAndInverses( ) . . . . . . . . . . . . . . . . . for groups ## DeclareOperation( "GeneratorsAndInverses", [ IsMagmaWithInverses ] ); ############################################################################# ## this function has been copied from main library gprd.gd and renamed ## #O SubdirectProductWithEmbeddings(, , , ) ## DeclareGlobalFunction("SubdirectProductWithEmbeddings"); DeclareOperation( "SubdirectProductWithEmbeddingsOp", [ IsGroup, IsGroup, IsGroupHomomorphism, IsGroupHomomorphism ] ); DeclareAttribute( "SubdirectProductWithEmbeddingsInfo", IsGroup, "mutable" ); ############################################################################# ## #E groups.gd . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/start.gd000644 000765 000024 00000004552 14621444241 020142 0ustar00christopherwensleystaff000000 000000 ############################################################################## ## #W start.gd GAP4 package `Utils' Chris Wensley ## #Y Copyright (C) 2015-2022, The GAP Group DeclareInfoClass( "InfoUtils" ); ## these version numbers refer to the latest package releases ## containing the code which is to be transferred UtilsPackageVersions := [ "autodoc", "2016.01.31", ## latest is 2017.09.08 "resclasses", "4.2.5", ## latest is 4.7.1, 18/12/17 "rcwa", "4.1.5" ## latest is 4.6.2, 26/06/18 ]; ## (04/09/18) removed QPA from this list - no functions transferred ## (02/06/18) removed xmod from this list ## functions being transferred cannot be used until the Home packages have ## been loaded: when this has been done the following variable is set true UtilsLoadingComplete := false; ############################################################################# ## #F OKtoReadFromUtils( ) . . . tests whether name still contains the ## code to be transferred to Utils #F OKtoReadFromUtilsSpec( , ) . . . special case of above ## BindGlobal( "OKtoReadFromUtils", function( Name ) local name, ver, ver0, pos, ok; name := LowercaseString( Name ); pos := Position( UtilsPackageVersions, name ); if ( pos = fail ) then Error( "package 'name' not in the list UtilsPackageVersions" ); fi; ver0 := UtilsPackageVersions[ pos+1 ]; ver := InstalledPackageVersion( name ); ok := ( ( ver = fail ) ## name is not installed on the system or ( ver > ver0 ) ); ## name still contains the code return ok; end ); BindGlobal( "OKtoReadFromUtilsSpec", function( Name, oldver ) local name, ver, pos, ok; name := LowercaseString( Name ); pos := Position( UtilsPackageVersions, name ); if ( pos = fail ) then Error( "package 'name' not in the list UtilsPackageVersions" ); fi; ver := InstalledPackageVersion( name ); ok := ( ( ver = fail ) ## name is not installed on the system or ( ver > oldver ) ); ## name still contains the code return ok; end ); ############################################################################# ## #E start.gd . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/string.gi000644 000765 000024 00000001560 14621444241 020314 0ustar00christopherwensleystaff000000 000000 ############################################################################# ## #W string.gi GAP4 package `Utils' Sebastian Gutsche ## Stefan Kohl ## #Y Copyright (C) 2015-2018, The GAP Group ############################################################################# ## this function has been transferred from ResClasses ## #F BlankFreeString( ) . . . . . . . . . . . . . string without blanks ## InstallGlobalFunction( BlankFreeString, function ( obj ) local str; str := String(obj); RemoveCharacters(str," "); return str; end ); ############################################################################# ## #E string.gi . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/latex.gi000644 000765 000024 00000003224 14621444241 020122 0ustar00christopherwensleystaff000000 000000 ############################################################################# ## #W latex.gi GAP4 package `Utils' Stefan Kohl ## #Y Copyright (C) 2015-2016, The GAP Group ############################################################################# ## this function has been transferred from ResClasses ## #F IntOrInfinityToLaTeX( n ) . LaTeX string for a given integer or infinity ## InstallGlobalFunction( IntOrInfinityToLaTeX, function( n ) if IsInt(n) then return String(n); elif IsInfinity(n) then return "\\infty"; else return fail; fi; end ); ############################################################################# ## this function has been transferred from RCWA ## #F LaTeXStringFactorsInt( ) . . . . prime factorization in LaTeX format ## InstallGlobalFunction( LaTeXStringFactorsInt, function ( n ) local facts, str, i; if not IsInt(n) then Error("usage: LaTeXStringFactorsInt( ) for an integer "); fi; if n < 0 then str := "-"; n := -n; else str := ""; fi; facts := Collected(Factors(n)); for i in [1..Length(facts)] do Append(str,String(facts[i][1])); if facts[i][2] > 1 then Append(str,"^"); if facts[i][2] >= 10 then Append(str,"{"); fi; Append(str,String(facts[i][2])); if facts[i][2] >= 10 then Append(str,"}"); fi; fi; if i < Length(facts) then Append(str," \\cdot "); fi; od; return str; end ); ############################################################################# ## #E latex.gi . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/number.gd000644 000765 000024 00000004740 14621444241 020274 0ustar00christopherwensleystaff000000 000000 ############################################################################## ## #W number.gd GAP4 package `Utils' Stefan Kohl ## #Y Copyright (C) 2015-2018, The GAP Group ############################################################################# ## this function has been transferred from RCWA ## #F NextProbablyPrimeInt( ) . . next integer passing `IsProbablyPrimeInt' ## ## Returns the smallest integer larger than which passes GAP's ## probabilistic primality test. ## ## The function `NextProbablyPrimeInt' does the same as `NextPrimeInt', ## except that for reasons of performance it tests numbers only for ## `IsProbablyPrimeInt' instead of `IsPrimeInt'. ## For large , this function is much faster than `NextPrimeInt'. ## DeclareGlobalFunction( "NextProbablyPrimeInt" ); ############################################################################# ## this function has been transferred from RCWA ## #F AllSmoothIntegers( , ) ## ## Returns the set of all integers in the range [1..] which have only ## prime divisors in the range [2..]. ## DeclareGlobalFunction( "AllSmoothIntegers" ); ############################################################################# ## this function has been transferred from RCWA ## #F PrimeNumbersIterator( ) #F PrimeNumbersIterator( chunksize ) ## ## Iterator running through the prime numbers in ascending order. ## DeclareGlobalFunction ( "PrimeNumbersIterator" ); ############################################################################# ## this function has been transferred from RCWA ## #F RestrictedPartitionsWithoutRepetitions( , ) ## ## Given a positive integer n and a set of positive integers S, this func- ## tion returns a list of all partitions of n into distinct elements of S. ## The only difference to `RestrictedPartitions' is that no repetitions are ## allowed. ## DeclareGlobalFunction( "RestrictedPartitionsWithoutRepetitions" ); ############################################################################# ## this function has been transferred from RCWA ## #O AllProducts( , ) . . all products of -tuples of elements of #M AllProducts( , ) . . . . . . . . . . . . . . . . . . . . for lists ## DeclareOperation( "AllProducts", [ IsListOrCollection, IsPosInt ] ); ############################################################################# ## #E number.gd . . . . . . . . . . . . . . . . . . . . . . . . . . . ends here utils-0.86/lib/download.gd000644 000765 000024 00000002243 14621444241 020607 0ustar00christopherwensleystaff000000 000000 ############################################################################## ## #W download.gd GAP4 package `Utils' Thomas Breuer ## #Y Copyright (C) 2022, The GAP Group ############################################################################# ## This function is intended to be used instead of similar ones from ## various packages (AtlasRep, FactInt, GAPDoc, PackageManager, ## StandardFF, ...) ## #O Download( [, ] ) ## DeclareOperation( "Download", [ IsString ] ); DeclareOperation( "Download", [ IsString, IsRecord ] ); ############################################################################# ## #U DownloadVerifyCertificate ## DeclareUserPreference( rec( name:= "DownloadVerifyCertificate", description:= [ "The value 'true' (the default) means that the server's certificate \ is checked in calls of 'Download' such that nothing gets downloaded \ if the certificate is invalid. \ If the value is 'false' then those download methods that are based on \ curl or wget will omit the check of the server's certificate." ], default:= true, values:= [ true, false ], multi:= false, package:= "utils", ) );