congruence-1.2.6/ 000755 000766 000024 00000000000 14577601110 013652 5 ustar 00obk1 staff 000000 000000 congruence-1.2.6/PackageInfo.g 000644 000766 000024 00000011351 14577601060 016176 0 ustar 00obk1 staff 000000 000000 #############################################################################
##
#W PackageInfo.g The Congruence package Ann Dooms
#W Eric Jespers
#W Olexandr Konovalov
#W Helena Verrill
##
##
#############################################################################
SetPackageInfo( rec(
PackageName := "Congruence",
Subtitle := "Congruence subgroups of SL(2,Integers)",
Version := "1.2.6",
Date := "23/03/2024", # dd/mm/yyyy format
License := "GPL-2.0-or-later",
## <#GAPDoc Label="PKGVERSIONDATA">
##
##
##
## <#/GAPDoc>
SourceRepository := rec(
Type := "git",
URL := Concatenation( "https://github.com/gap-packages/", LowercaseString(~.PackageName) ),
),
IssueTrackerURL := Concatenation( ~.SourceRepository.URL, "/issues" ),
PackageWWWHome := Concatenation( "https://gap-packages.github.io/", LowercaseString(~.PackageName) ),
README_URL := Concatenation( ~.PackageWWWHome, "/README.md" ),
PackageInfoURL := Concatenation( ~.PackageWWWHome, "/PackageInfo.g" ),
ArchiveURL := Concatenation( ~.SourceRepository.URL,
"/releases/download/v", ~.Version,
"/", LowercaseString(~.PackageName), "-", ~.Version ),
ArchiveFormats := ".tar.gz",
Persons := [
rec(
LastName := "Dooms",
FirstNames := "Ann",
IsAuthor := true,
IsMaintainer := true,
Email := "andooms@vub.ac.be",
WWWHome := "http://homepages.vub.ac.be/~andooms",
PostalAddress := Concatenation( [
"Department of Mathematics\n",
"Vrije Universiteit Brussel\n",
"Pleinlaan 2, Brussels, B-1050 Belgium" ] ),
Place := "Brussels",
Institution := "Vrije Universiteit Brussel"
),
rec(
LastName := "Jespers",
FirstNames := "Eric",
IsAuthor := true,
IsMaintainer := false,
Email := "efjesper@vub.ac.be",
WWWHome := "http://homepages.vub.ac.be/~efjesper",
PostalAddress := Concatenation( [
"Department of Mathematics\n",
"Vrije Universiteit Brussel\n",
"Pleinlaan 2, Brussels, B-1050 Belgium" ] ),
Place := "Brussels",
Institution := "Vrije Universiteit Brussel"
),
rec(
LastName := "Konovalov",
FirstNames := "Olexandr",
IsAuthor := true,
IsMaintainer := true,
Email := "obk1@st-andrews.ac.uk",
WWWHome := "https://olexandr-konovalov.github.io/",
PostalAddress := Concatenation( [
"School of Computer Science\n",
"University of St Andrews\n",
"Jack Cole Building, North Haugh,\n",
"St Andrews, Fife, KY16 9SX, Scotland" ] ),
Place := "St Andrews",
Institution := "University of St Andrews"
),
rec(
LastName := "Verrill",
FirstNames := "Helena",
IsAuthor := true,
IsMaintainer := true,
Email := "verrill@math.lsu.edu",
WWWHome := "http://www.math.lsu.edu/~verrill",
PostalAddress := Concatenation( [
"Department of Mathematics\n",
"Louisiana State University\n",
"Baton Rouge, Louisiana, 70803-4918\n",
"USA" ] ),
Place := "Baton Rouge",
Institution := "Louisiana State University"
)
],
Status := "accepted",
CommunicatedBy := "Graham Ellis (Galway)",
AcceptDate := "09/2014",
AbstractHTML := "The Congruence package provides functions to construct several types of canonical congruence subgroups in SL_2(Z), and also intersections of a finite number of such subgroups. Furthermore, it implements the algorithm for generating Farey symbols for congruence subgroups and using them to produce a system of independent generators for these subgroups",
PackageDoc := rec(
BookName := "Congruence",
ArchiveURLSubset := ["doc"],
HTMLStart := "doc/chap0_mj.html",
PDFFile := "doc/manual.pdf",
SixFile := "doc/manual.six",
LongTitle := "Congruence subgroups of SL(2,Integers)",
Autoload := true
),
Dependencies := rec(
GAP := ">=4.8",
NeededOtherPackages := [ ["GAPDoc", ">= 1.5.1"] ],
SuggestedOtherPackages := [],
ExternalConditions := []
),
AvailabilityTest := ReturnTrue,
TestFile := "tst/testall.g",
Keywords := ["congruence subgroup", "Farey symbol"]
));
congruence-1.2.6/README.md 000644 000766 000024 00000002144 14577601060 015136 0 ustar 00obk1 staff 000000 000000 [](https://github.com/gap-packages/congruence/actions?query=workflow%3ACI+branch%3Amaster)
[](https://codecov.io/gh/gap-packages/congruence)
# GAP package Congruence
The GAP package Congruence provides functions to construct several types of
canonical congruence subgroups in SL_2(Z), and also intersections of a
finite number of such subgroups.
Furthermore, it implements the algorithm for generating Farey symbols for
congruence subgroups and using them to produce a system of independent
generators for these subgroups.
Congruence does not use external binaries and, therefore, works without
restrictions on the type of the operating system. It is redistributed with
GAP, but is not loaded by default. Therefore, to use Congruence, first you
need to load it using the following command:
gap> LoadPackage("congruence");
Ann Dooms, Eric Jespers, Olexandr Konovalov, Helena Verrill
congruence-1.2.6/COPYING 000644 000766 000024 00000043110 14577601060 014710 0 ustar 00obk1 staff 000000 000000 GNU GENERAL PUBLIC LICENSE
Version 2, June 1991
Copyright (C) 1989, 1991 Free Software Foundation, Inc.
59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
Everyone is permitted to copy and distribute verbatim copies
of this license document, but changing it is not allowed.
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congruence-1.2.6/makedoc.g 000644 000766 000024 00000007477 14577601060 015450 0 ustar 00obk1 staff 000000 000000 ###########################################################################
##
#W makedoc.g The Congruence package Olexandr Konovalov
##
###########################################################################
ExtractMyManualExamples:=function( pkgname, main, files )
local path, tst, i, s, basename, name, output, ch, a, location, pos, comment;
path:=Directory(
Concatenation(PackageInfo(pkgname)[1].InstallationPath, "/doc") );
Print("Extracting manual examples for ", pkgname, " package ...\n" );
tst:=ExtractExamples( path, main, files, "Chapter" );
Print(Length(tst), " chapters detected\n");
for i in [ 1 .. Length(tst) ] do
Print( "Chapter ", i, " : \c" );
if Length( tst[i] ) > 0 then
s := String(i);
if Length(s)=1 then
# works for <100 chapters
s:=Concatenation("0",s);
fi;
basename:=Concatenation( LowercaseString(pkgname), s, ".tst" );
name := Filename(
Directory(
Concatenation( PackageInfo(pkgname)[1].InstallationPath,
"/tst" ) ), basename );
output := OutputTextFile( name, false ); # to empty the file first
SetPrintFormattingStatus( output, false ); # to avoid line breaks
ch := tst[i];
AppendTo(output, "# ", pkgname, ", chapter ",i,"\n");
AppendTo(output, "#\n",
"# DO NOT EDIT THIS FILE - EDIT EXAMPLES IN THE SOURCE INSTEAD!\n",
"#\n",
"# This file has been autogenerated with GAP. It contains examples\n",
"# extracted from the documentation. Each example is preceded by the\n",
"# comment which points to the location of its source.\n",
"#\n");
AppendTo(output, "gap> START_TEST( \"", basename, "\");\n\n");
for a in ch do
location := a[2][1];
pos := PositionSublist(LowercaseString(location),LowercaseString(pkgname));
if pos <> fail then
comment := location{[ pos+Length(pkgname)+1 .. Length(location) ]};
else
pos := PositionSublist(location,".//");
comment := location{[ pos+3 .. Length(location) ]};
fi;
AppendTo(output, "# ", comment, ":", a[2][2], "-", a[2][3], a[1]);
od;
AppendTo(output, "gap> STOP_TEST(\"", basename, "\", 1 );\n");
Print("extracted ", Length(ch), " examples \n");
else
Print("no examples \n" );
fi;
od;
end;
###########################################################################
CONGRUENCEMANUALFILES:=[
"../PackageInfo.g"
];
MakeGAPDocDoc( "doc", # path to the directory containing the main file
"manual", # the name of the main file (without extension)
# list of (probably source code) files relative
# to path which contain pieces of documentation
# which must be included in the document
CONGRUENCEMANUALFILES,
"Congruence",# the name of the book used by GAP's online help
"../../..",# optional: relative path to the main GAP root
# directory to produce HTML files with relative
# paths to external books.
"MathJax" # optional: use "MathJax", "Tth" and/or "MathML"
# to produce additional variants of HTML files
);;
# Copy the *.css and *.js files from the styles directory of the GAPDoc
# package into the directory containing the package manual.
CopyHTMLStyleFiles( "doc" );
# Create the manual.lab file which is needed if the main manuals or another
# package is referring to your package
GAPDocManualLab( "Congruence" );;
ExtractMyManualExamples( "congruence", "manual.xml", CONGRUENCEMANUALFILES );
QUIT;
###########################################################################
##
#E
##
congruence-1.2.6/lib/ 000755 000766 000024 00000000000 14577601060 014424 5 ustar 00obk1 staff 000000 000000 congruence-1.2.6/doc/ 000755 000766 000024 00000000000 14577601110 014417 5 ustar 00obk1 staff 000000 000000 congruence-1.2.6/tst/ 000755 000766 000024 00000000000 14577601060 014470 5 ustar 00obk1 staff 000000 000000 congruence-1.2.6/init.g 000644 000766 000024 00000001474 14577601060 014777 0 ustar 00obk1 staff 000000 000000 #############################################################################
##
#W init.g The Congruence package Ann Dooms
#W Eric Jespers
#W Olexandr Konovalov
#W Helena Verrill
##
##
#############################################################################
# read Congruence declarations
ReadPackage( "congruence", "lib/cong.gd" );
ReadPackage( "congruence", "lib/farey.gd" );
# read the other part of code
ReadPackage( "congruence", "lib/cong.g" );
ReadPackage( "congruence", "lib/buildman.g" );
ReadPackage( "congruence", "lib/factor.g" );
# set the default InfoLevel
SetInfoLevel( InfoCongruence, 1 );
congruence-1.2.6/read.g 000644 000766 000024 00000001236 14577601060 014743 0 ustar 00obk1 staff 000000 000000 #############################################################################
##
#W read.g The Congruence package Ann Dooms
#W Eric Jespers
#W Olexandr Konovalov
#W Helena Verrill
##
##
#############################################################################
# read the implementation part of the Congruence package
ReadPackage( "congruence", "lib/cong.gi" );
ReadPackage( "congruence", "lib/farey.gi" );
ReadPackage( "congruence", "lib/random.gi" );
congruence-1.2.6/tst/congruence04.tst 000644 000766 000024 00000007634 14577601110 017526 0 ustar 00obk1 staff 000000 000000 # congruence, chapter 4
#
# DO NOT EDIT THIS FILE - EDIT EXAMPLES IN THE SOURCE INSTEAD!
#
# This file has been autogenerated with GAP. It contains examples
# extracted from the documentation. Each example is preceded by the
# comment which points to the location of its source.
#
gap> START_TEST( "congruence04.tst");
# doc/gens.xml:30-47
gap> FareySymbol(PrincipalCongruenceSubgroup(8));
[ infinity, 0, 1/4, 1/3, 3/8, 2/5, 1/2, 3/5, 5/8, 2/3, 3/4, 1, 5/4, 4/3,
11/8, 7/5, 3/2, 8/5, 13/8, 5/3, 7/4, 2, 9/4, 7/3, 19/8, 12/5, 5/2, 13/5,
21/8, 8/3, 11/4, 3, 13/4, 10/3, 27/8, 17/5, 7/2, 18/5, 29/8, 11/3, 15/4, 4,
17/4, 13/3, 9/2, 14/3, 19/4, 5, 21/4, 16/3, 11/2, 17/3, 23/4, 6, 25/4,
19/3, 13/2, 20/3, 27/4, 7, 29/4, 22/3, 15/2, 23/3, 31/4, 8, infinity ]
[ 1, 17, 10, 26, 32, 18, 19, 27, 30, 5, 2, 2, 13, 28, 26, 20, 21, 29, 27, 7,
3, 3, 16, 31, 28, 22, 23, 33, 29, 9, 4, 4, 5, 30, 31, 24, 25, 32, 33, 12,
6, 6, 7, 19, 18, 15, 8, 8, 9, 21, 20, 10, 11, 11, 12, 23, 22, 13, 14, 14,
15, 25, 24, 16, 17, 1 ]
gap> FareySymbol(CongruenceSubgroupGamma0(20));
[ infinity, 0, 1/5, 1/4, 2/7, 3/10, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1,
infinity ]
[ 1, 3, 4, 6, 7, 7, 5, 2, 2, 3, 6, 4, 5, 1 ]
# doc/gens.xml:128-140
gap> H:=CongruenceSubgroupGamma0(5);
gap> fs:=FareySymbol(H);
[ infinity, 0, 1/2, 1, infinity ]
[ 1, "even", "even", 1 ]
gap> gfs:=GeneralizedFareySequence(fs);
[ infinity, 0, 1/2, 1, infinity ]
gap> MatrixByEvenInterval(gfs,2);
[ [ 2, -1 ], [ 5, -2 ] ]
# doc/gens.xml:152-160
gap> fs_oo:=FareySymbolByData([infinity,0,infinity],["odd","odd"]);;
gap> gfs_oo:=GeneralizedFareySequence(fs_oo);
[ infinity, 0, infinity ]
gap> MatrixByOddInterval(gfs_oo,1);
[ [ -1, -1 ], [ 1, 0 ] ]
# doc/gens.xml:172-179
gap> fs_free:=FareySymbolByData([infinity,0,1,2,infinity],[1,2,2,1]);;
gap> gfs_free:=GeneralizedFareySequence(fs_free);;
gap> MatrixByFreePairOfIntervals(gfs_free,2,3);
[ [ 3, -2 ], [ 2, -1 ] ]
# doc/gens.xml:191-203
gap> fs_eo:=FareySymbolByData([infinity,0,infinity],["even","odd"]);;
gap> GeneratorsByFareySymbol(last);
[ [ [ 0, -1 ], [ 1, 0 ] ], [ [ 0, -1 ], [ 1, -1 ] ] ]
gap> GeneratorsByFareySymbol(fs);
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 2, -1 ], [ 5, -2 ] ], [ [ 3, -2 ], [ 5, -3 ] ] ]
gap> GeneratorsByFareySymbol(fs_oo);
[ [ [ -1, -1 ], [ 1, 0 ] ], [ [ 0, -1 ], [ 1, -1 ] ] ]
gap> GeneratorsByFareySymbol(fs_free);
[ [ [ 1, 2 ], [ 0, 1 ] ], [ [ 3, -2 ], [ 2, -1 ] ] ]
# doc/gens.xml:215-241
gap> G:=PrincipalCongruenceSubgroup(2);
gap> FareySymbol(G);
[ infinity, 0, 1, 2, infinity ]
[ 2, 1, 1, 2 ]
gap> GeneratorsOfGroup(G);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 2 ], [ 0, 1 ] ], [ [ 3, -2 ], [ 2, -1 ] ] ]
gap> H:=CongruenceSubgroupGamma0(5);
gap> GeneratorsOfGroup(H);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 2, -1 ], [ 5, -2 ] ], [ [ 3, -2 ], [ 5, -3 ] ] ]
gap> I:=IntersectionOfCongruenceSubgroups(PrincipalCongruenceSubgroup(2),CongruenceSubgroupGamma0(3));
gap> FareySymbol(I);
[ infinity, 0, 1/3, 1/2, 2/3, 1, 4/3, 3/2, 5/3, 2, infinity ]
[ 1, 5, 4, 3, 2, 2, 3, 4, 5, 1 ]
gap> GeneratorsOfGroup(I);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 2 ], [ 0, 1 ] ], [ [ 11, -2 ], [ 6, -1 ] ],
[ [ 19, -8 ], [ 12, -5 ] ], [ [ 17, -10 ], [ 12, -7 ] ],
[ [ 7, -6 ], [ 6, -5 ] ] ]
# doc/gens.xml:262-271
gap> IndexInPSL2ZByFareySymbol(fs);
6
gap> IndexInPSL2ZByFareySymbol(fs_oo);
2
gap> IndexInPSL2ZByFareySymbol(fs_free);
6
gap> STOP_TEST("congruence04.tst", 1 );
congruence-1.2.6/tst/congruence03.tst 000644 000766 000024 00000001750 14577601110 017516 0 ustar 00obk1 staff 000000 000000 # congruence, chapter 3
#
# DO NOT EDIT THIS FILE - EDIT EXAMPLES IN THE SOURCE INSTEAD!
#
# This file has been autogenerated with GAP. It contains examples
# extracted from the documentation. Each example is preceded by the
# comment which points to the location of its source.
#
gap> START_TEST( "congruence03.tst");
# doc/farey.xml:59-65
gap> fs:=FareySymbolByData([infinity,0,1,2,infinity],[1,2,2,1]);
[ infinity, 0, 1, 2, infinity ]
[ 1, 2, 2, 1 ]
# doc/farey.xml:76-81
gap> IsValidFareySymbol(fs);
true
# doc/farey.xml:99-104
gap> GeneralizedFareySequence(fs);
[ infinity, 0, 1, 2, infinity ]
# doc/farey.xml:121-126
gap> List([1..5], i -> NumeratorOfGFSElement(GeneralizedFareySequence(fs),i));
[ -1, 0, 1, 2, 1 ]
# doc/farey.xml:142-147
gap> List([1..5], i -> DenominatorOfGFSElement(GeneralizedFareySequence(fs),i));
[ 0, 1, 1, 1, 0 ]
# doc/farey.xml:159-164
gap> LabelsOfFareySymbol(fs);
[ 1, 2, 2, 1 ]
gap> STOP_TEST("congruence03.tst", 1 );
congruence-1.2.6/tst/congruence02.tst 000644 000766 000024 00000007215 14577601110 017517 0 ustar 00obk1 staff 000000 000000 # congruence, chapter 2
#
# DO NOT EDIT THIS FILE - EDIT EXAMPLES IN THE SOURCE INSTEAD!
#
# This file has been autogenerated with GAP. It contains examples
# extracted from the documentation. Each example is preceded by the
# comment which points to the location of its source.
#
gap> START_TEST( "congruence02.tst");
# doc/cong.xml:71-94
gap> G_8:=PrincipalCongruenceSubgroup(8);
gap> IsGroup(G_8);
true
gap> IsMatrixGroup(G_8);
true
gap> DimensionOfMatrixGroup(G_8);
2
gap> MultiplicativeNeutralElement(G_8);
[ [ 1, 0 ], [ 0, 1 ] ]
gap> One(G);
[ [ 1, 0 ], [ 0, 1 ] ]
gap> [[1,2],[3,4]] in G_8;
false
gap> [[1,8],[8,65]] in G_8;
true
gap> SL_2:=SL(2,Integers);
SL(2,Integers)
gap> IsSubgroup(SL_2,G_8);
true
# doc/cong.xml:128-133
gap> G0_4:=CongruenceSubgroupGamma0(4);
# doc/cong.xml:167-172
gap> GU0_2:=CongruenceSubgroupGammaUpper0(2);
# doc/cong.xml:206-211
gap> G1_6:=CongruenceSubgroupGamma1(6);
# doc/cong.xml:245-250
gap> GU1_4:=CongruenceSubgroupGammaUpper1(4);
# doc/cong.xml:279-286
gap> I:=IntersectionOfCongruenceSubgroups(G0_4,GU1_4);
gap> J:=IntersectionOfCongruenceSubgroups(G0_4,G1_6);
# doc/cong.xml:313-322
gap> IsPrincipalCongruenceSubgroup(G_8);
true
gap> IsPrincipalCongruenceSubgroup(G0_4);
false
gap> IsPrincipalCongruenceSubgroup(I);
true
# doc/cong.xml:386-393
gap> IsIntersectionOfCongruenceSubgroups(I);
false
gap> IsIntersectionOfCongruenceSubgroups(J);
true
# doc/cong.xml:413-424
gap> LevelOfCongruenceSubgroup(G_8);
8
gap> LevelOfCongruenceSubgroup(G1_6);
6
gap> LevelOfCongruenceSubgroup(I);
4
gap> LevelOfCongruenceSubgroup(J);
12
# doc/cong.xml:435-446
gap> IndexInSL2Z(G_8);
384
gap> G_2:=PrincipalCongruenceSubgroup(2);
gap> IndexInSL2Z(G_2);
12
gap> IndexInSL2Z(GU1_4);
12
# doc/cong.xml:462-476
gap> DefiningCongruenceSubgroups(J);
[ ,
]
gap> P:=PrincipalCongruenceSubgroup(6);
gap> Q:=PrincipalCongruenceSubgroup(10);
gap> G:=IntersectionOfCongruenceSubgroups(Q,P);
gap> DefiningCongruenceSubgroups(G);
[ ]
# doc/cong.xml:503-510
gap> Random(G_2) in G_2;
true
gap> Random(G_8,2) in G_8;
true
# doc/cong.xml:522-529
gap> \in([ [ 21, 10 ], [ 2, 1 ] ],G_2);
true
gap> \in([ [ 21, 10 ], [ 2, 1 ] ],G_8);
false
# doc/cong.xml:551-556
gap> CanEasilyCompareCongruenceSubgroups(G_8,I);
false
# doc/cong.xml:570-588
gap> IsSubset(G_2,G_8);
true
gap> IsSubset(G_8,G_2);
false
gap> f:=[PrincipalCongruenceSubgroup,CongruenceSubgroupGamma1,CongruenceSubgroupGammaUpper1,CongruenceSubgroupGamma0,CongruenceSubgroupGammaUpper0];;
gap> g1:=List(f, t -> t(2));;
gap> g2:=List(f, t -> t(4));;
gap> for g in g2 do
> Print( List( g1, x -> IsSubgroup(x,g) ), "\n");
> od;
[ true, true, true, true, true ]
[ false, true, false, true, false ]
[ false, false, true, false, true ]
[ false, false, false, true, false ]
[ false, false, false, false, true ]
# doc/cong.xml:601-606
gap> Index(G_2,G_8);
32
gap> STOP_TEST("congruence02.tst", 1 );
congruence-1.2.6/tst/testall.g 000644 000766 000024 00000000373 14577601060 016313 0 ustar 00obk1 staff 000000 000000 LoadPackage( "congruence" );
TestDirectory(DirectoriesPackageLibrary( "congruence", "tst" ),
rec(exitGAP := true,
testOptions := rec(compareFunction := "uptowhitespace") ) );
FORCE_QUIT_GAP(1); # if we ever get here, there was an error
congruence-1.2.6/tst/cong.tst 000644 000766 000024 00000053247 14577601060 016165 0 ustar 00obk1 staff 000000 000000 gap> G:=PrincipalCongruenceSubgroup(8);
gap> IsGroup(G);
true
gap> IsMatrixGroup(G);
true
gap> IsPrincipalCongruenceSubgroup(G);
true
gap> IsFinitelyGeneratedGroup(G);
true
gap> LevelOfCongruenceSubgroup(G);
8
gap> DimensionOfMatrixGroup(G);
2
gap> MultiplicativeNeutralElement(G);
[ [ 1, 0 ], [ 0, 1 ] ]
gap> One(G);
[ [ 1, 0 ], [ 0, 1 ] ]
gap> [[1,2],[3,4]] in G;
false
gap> [[1,8],[8,65]] in G;
true
gap> G:=PrincipalCongruenceSubgroup(3);
gap> ForAll( List([1..100], k -> Random(G)), m -> m in G);
true
gap> ForAll( List([1..100], k -> Random(G,10*k)), m -> m in G);
true
gap> G:=CongruenceSubgroupGamma0(3);
gap> ForAll( List([1..100], k -> Random(G)), m -> m in G);
true
gap> ForAll( List([1..100], k -> Random(G,10*k)), m -> m in G);
true
gap> G:=CongruenceSubgroupGammaUpper0(3);
gap> ForAll( List([1..100], k -> Random(G)), m -> m in G);
true
gap> ForAll( List([1..100], k -> Random(G,10*k)), m -> m in G);
true
gap> G:=CongruenceSubgroupGamma1(3);
gap> ForAll( List([1..100], k -> Random(G)), m -> m in G);
true
gap> ForAll( List([1..100], k -> Random(G,10*k)), m -> m in G);
true
gap> G:=CongruenceSubgroupGammaUpper1(3);
gap> ForAll( List([1..100], k -> Random(G)), m -> m in G);
true
gap> ForAll( List([1..100], k -> Random(G,10*k)), m -> m in G);
true
gap> G2:=PrincipalCongruenceSubgroup(2);
gap> G3:=PrincipalCongruenceSubgroup(3);
gap> G6:=PrincipalCongruenceSubgroup(6);
gap> G:=SL(2,Integers);
SL(2,Integers)
gap> IsSubgroup(G,G2);
true
gap> IsSubgroup(G3,G2);
false
gap> IsSubgroup(G2,G6);
true
gap> Index(G,G3);
24
gap> IndexInSL2Z(G6);
144
gap> Index(G3,G6);
6
gap> f:=[PrincipalCongruenceSubgroup,
> CongruenceSubgroupGamma1,
> CongruenceSubgroupGammaUpper1,
> CongruenceSubgroupGamma0,
> CongruenceSubgroupGammaUpper0];;
gap> g1:=List(f, t -> t(2));;
gap> g2:=List(f, t -> t(4));;
gap> for g in g2 do
> Print( List( g1, x -> IsSubgroup(x,g) ), "\n");
> od;
[ true, true, true, true, true ]
[ false, true, false, true, false ]
[ false, false, true, false, true ]
[ false, false, false, true, false ]
[ false, false, false, false, true ]
gap> Intersection(G2,G3);
gap> G6=Intersection(G2,G3);
true
gap> g1:=List(f, t -> t(2));;
gap> g2:=List(f, t -> t(2));;
gap> for g in g2 do
> Print( List( g1, x -> Intersection(x,g) ), "\n");
> od;
[ PrincipalCongruenceSubgroup(2), PrincipalCongruenceSubgroup(2),
PrincipalCongruenceSubgroup(2), PrincipalCongruenceSubgroup(2),
PrincipalCongruenceSubgroup(2) ]
[ PrincipalCongruenceSubgroup(2), CongruenceSubgroupGamma1(2),
PrincipalCongruenceSubgroup(2), CongruenceSubgroupGamma1(2),
PrincipalCongruenceSubgroup(2) ]
[ PrincipalCongruenceSubgroup(2), PrincipalCongruenceSubgroup(2),
CongruenceSubgroupGammaUpper1(2), PrincipalCongruenceSubgroup(2),
CongruenceSubgroupGammaUpper1(2) ]
[ PrincipalCongruenceSubgroup(2), CongruenceSubgroupGamma1(2),
PrincipalCongruenceSubgroup(2), CongruenceSubgroupGamma0(2),
IntersectionOfCongruenceSubgroups(
CongruenceSubgroupGamma0(2),
CongruenceSubgroupGammaUpper0(2) ) ]
[ PrincipalCongruenceSubgroup(2), PrincipalCongruenceSubgroup(2),
CongruenceSubgroupGammaUpper1(2), IntersectionOfCongruenceSubgroups(
CongruenceSubgroupGamma0(2),
CongruenceSubgroupGammaUpper0(2) ), CongruenceSubgroupGammaUpper0(2) ]
gap> G:=Intersection(CongruenceSubgroupGamma0(4),CongruenceSubgroupGamma1(3));
gap> DefiningCongruenceSubgroups(G);
[ ,
]
gap> H:=Intersection(G,CongruenceSubgroupGamma1(4));
gap> DefiningCongruenceSubgroups(H);
[ ,
]
gap> K:=Intersection(H,CongruenceSubgroupGamma0(3));
gap> List([1..6], n -> IndexInSL2Z(PrincipalCongruenceSubgroup(n)));
[ 1, 12, 24, 48, 120, 144 ]
gap> fs:=FareySymbolByData([infinity,0,1,2,infinity],[1,2,2,1]);
[ infinity, 0, 1, 2, infinity ]
[ 1, 2, 2, 1 ]
gap> GeneralizedFareySequence(fs);
[ infinity, 0, 1, 2, infinity ]
gap> List([1..5], i -> NumeratorOfGFSElement(GeneralizedFareySequence(fs),i));
[ -1, 0, 1, 2, 1 ]
gap> List([1..5], i -> DenominatorOfGFSElement(GeneralizedFareySequence(fs),i));
[ 0, 1, 1, 1, 0 ]
gap> LabelsOfFareySymbol(fs);
[ 1, 2, 2, 1 ]
gap> IsValidFareySymbol(fs);
true
gap> fs:=FareySymbolByData([infinity,0,1,infinity],[1,"even",1]);
[ infinity, 0, 1, infinity ]
[ 1, "even", 1 ]
gap> Print(fs); Print("\n");
FareySymbolByData( [ infinity, 0, 1, infinity ], [ 1, "even", 1 ] )
gap> SetInfoLevel(InfoCongruence,1);
gap> fs1_1:=FareySymbolByData([infinity,0,infinity],["even","odd"]);;
gap> GeneratorsByFareySymbol(last);
[ [ [ 0, -1 ], [ 1, 0 ] ], [ [ 0, -1 ], [ 1, -1 ] ] ]
gap> fs2_1:=FareySymbolByData([infinity,0,infinity],["odd","odd"]);;
gap> GeneratorsByFareySymbol(last);
[ [ [ -1, -1 ], [ 1, 0 ] ], [ [ 0, -1 ], [ 1, -1 ] ] ]
gap> fs2_2:=FareySymbolByData([infinity,0,1,2,infinity],[1,2,2,1]);;
gap> GeneratorsByFareySymbol(last);
[ [ [ 1, 2 ], [ 0, 1 ] ], [ [ 3, -2 ], [ 2, -1 ] ] ]
gap> fs2_3:=FareySymbolByData([infinity,0,1,infinity],[1,"even",1]);;
gap> GeneratorsByFareySymbol(last);
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 1, -1 ], [ 2, -1 ] ] ]
gap> fs3_1:=FareySymbolByData([infinity,0,1,infinity],["even","even","even"]);;
gap> GeneratorsByFareySymbol(last);
[ [ [ 0, -1 ], [ 1, 0 ] ], [ [ 1, -1 ], [ 2, -1 ] ], [ [ 1, -2 ], [ 1, -1 ] ]
]
gap> fs3_2:=FareySymbolByData([infinity,0,1,2,infinity],["even",1,"even",1]);;
gap> GeneratorsByFareySymbol(last);
[ [ [ 0, -1 ], [ 1, 0 ] ], [ [ 3, -1 ], [ 1, 0 ] ], [ [ 3, -5 ], [ 2, -3 ] ] ]
gap> fs3_3:=FareySymbolByData([infinity,0,1,2,5/2,3,infinity],[1,2,3,3,2,1]);;
gap> GeneratorsByFareySymbol(last);
[ [ [ 1, 3 ], [ 0, 1 ] ], [ [ 8, -3 ], [ 3, -1 ] ], [ [ 7, -12 ], [ 3, -5 ] ]
]
gap> fs3_4:=FareySymbolByData([infinity,0,1,infinity],[1,"odd",1]);;
gap> GeneratorsByFareySymbol(last);
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 1, -1 ], [ 3, -2 ] ] ]
gap> fs4_1:=FareySymbolByData([infinity,0,1/2,1,3/2,2,infinity],[1,2,3,3,2,1]);;
gap> GeneratorsByFareySymbol(last);
[ [ [ 1, 2 ], [ 0, 1 ] ], [ [ 7, -2 ], [ 4, -1 ] ], [ [ 5, -4 ], [ 4, -3 ] ] ]
gap> fs4_2:=FareySymbolByData([infinity,0,1/2,1,3/2,2,5/2,3,7/2,4,infinity],[1,4,5,5,3,3,2,2,4,1]);;
gap> GeneratorsByFareySymbol(last);
[ [ [ 1, 4 ], [ 0, 1 ] ], [ [ 15, -4 ], [ 4, -1 ] ], [ [ 5, -4 ], [ 4, -3 ] ],
[ [ 9, -16 ], [ 4, -7 ] ], [ [ 13, -36 ], [ 4, -11 ] ] ]
gap> fs4_4:=FareySymbolByData([infinity,0,1/2,1,infinity],[1,2,2,1]);;
gap> GeneratorsByFareySymbol(last);
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 3, -1 ], [ 4, -1 ] ] ]
gap> fs4_5:=FareySymbolByData([infinity,0,1,2,infinity],[1,"even","even",1]);;
gap> GeneratorsByFareySymbol(last);
[ [ [ 1, 2 ], [ 0, 1 ] ], [ [ 1, -1 ], [ 2, -1 ] ], [ [ 3, -5 ], [ 2, -3 ] ] ]
gap> fs4_6:=FareySymbolByData([infinity,0,1,2,3,4,infinity],[1,"even",2,"even",2,1]);;
gap> GeneratorsByFareySymbol(last);
[ [ [ 1, 4 ], [ 0, 1 ] ], [ [ 1, -1 ], [ 2, -1 ] ], [ [ 7, -11 ], [ 2, -3 ] ],
[ [ 5, -13 ], [ 2, -5 ] ] ]
gap> fs4_7:=FareySymbolByData([infinity,0,1,infinity],["odd","even","even"]);;
gap> GeneratorsByFareySymbol(last);
[ [ [ -1, -1 ], [ 1, 0 ] ], [ [ 1, -1 ], [ 2, -1 ] ],
[ [ 1, -2 ], [ 1, -1 ] ] ]
gap> fs5_1:=FareySymbolByData([infinity,0,1,2,3,infinity],[1,"even","even",1,"odd"]);;
gap> GeneratorsByFareySymbol(last);
[ [ [ 3, 2 ], [ 1, 1 ] ], [ [ 1, -1 ], [ 2, -1 ] ], [ [ 3, -5 ], [ 2, -3 ] ],
[ [ 3, -13 ], [ 1, -4 ] ] ]
gap> fs5_2:=FareySymbolByData([infinity,0,1/2,1,4/3,3/2,2,3,infinity],[1,2,3,3,"odd",2,1,"odd"]);;
gap> GeneratorsByFareySymbol(last);
[ [ [ 3, 2 ], [ 1, 1 ] ], [ [ 7, -2 ], [ 4, -1 ] ], [ [ 6, -5 ], [ 5, -4 ] ],
[ [ 26, -37 ], [ 19, -27 ] ], [ [ 3, -13 ], [ 1, -4 ] ] ]
gap> fs5_3:=FareySymbolByData([infinity,0,1/2,1,4/3,7/5,3/2,2,7/3,12/5,5/2,3,10/3,17/5,7/2,4,
> 13/3,22/5,9/2,23/5,14/3,5,infinity],
> [1,2,6,6,10,11,5,5,11,9,4,4,9,8,3,3,8,7,7,10,2,1]);
[ infinity, 0, 1/2, 1, 4/3, 7/5, 3/2, 2, 7/3, 12/5, 5/2, 3, 10/3, 17/5, 7/2,
4, 13/3, 22/5, 9/2, 23/5, 14/3, 5, infinity ]
[ 1, 2, 6, 6, 10, 11, 5, 5, 11, 9, 4, 4, 9, 8, 3, 3, 8, 7, 7, 10, 2, 1 ]
gap> GeneratorsByFareySymbol(fs5_3);
[ [ [ 1, 5 ], [ 0, 1 ] ], [ [ 24, -5 ], [ 5, -1 ] ], [ [ 6, -5 ], [ 5, -4 ] ],
[ [ 139, -190 ], [ 30, -41 ] ], [ [ 59, -85 ], [ 25, -36 ] ],
[ [ 11, -20 ], [ 5, -9 ] ], [ [ 84, -205 ], [ 25, -61 ] ],
[ [ 16, -45 ], [ 5, -14 ] ], [ [ 109, -375 ], [ 25, -86 ] ],
[ [ 21, -80 ], [ 5, -19 ] ], [ [ 91, -405 ], [ 20, -89 ] ] ]
gap> fs11_1:=FareySymbolByData([infinity,-1,0,1,2,infinity],["even","odd","odd","even","even"]);
[ infinity, -1, 0, 1, 2, infinity ]
[ "even", "odd", "odd", "even", "even" ]
gap> GeneratorsByFareySymbol(last);
[ [ [ -1, -2 ], [ 1, 1 ] ], [ [ -2, -1 ], [ 3, 1 ] ],
[ [ 1, -1 ], [ 3, -2 ] ], [ [ 3, -5 ], [ 2, -3 ] ],
[ [ 2, -5 ], [ 1, -2 ] ] ]
gap> fs11_2:=FareySymbolByData([infinity,0,1,2,3,infinity],["even","even","odd","odd","even"]);
[ infinity, 0, 1, 2, 3, infinity ]
[ "even", "even", "odd", "odd", "even" ]
gap> GeneratorsByFareySymbol(last);
[ [ [ 0, -1 ], [ 1, 0 ] ], [ [ 1, -1 ], [ 2, -1 ] ], [ [ 4, -7 ], [ 3, -5 ] ],
[ [ 7, -19 ], [ 3, -8 ] ], [ [ 3, -10 ], [ 1, -3 ] ] ]
gap> fs12_1:=FareySymbolByData([infinity,0,1/6,1/5,1/4,1/3,1/2,3/5,2/3,1,4/3,7/5,3/2,5/3,7/4,9/5,11/6,2,infinity],
> [ 1,8, 9, 7, 4, 4, 7, 6, 2,2, 6, 5, 3, 3, 5, 9, 8,1]);
[ infinity, 0, 1/6, 1/5, 1/4, 1/3, 1/2, 3/5, 2/3, 1, 4/3, 7/5, 3/2, 5/3, 7/4,
9/5, 11/6, 2, infinity ]
[ 1, 8, 9, 7, 4, 4, 7, 6, 2, 2, 6, 5, 3, 3, 5, 9, 8, 1 ]
gap> GeneratorsByFareySymbol(last);
[ [ [ 1, 2 ], [ 0, 1 ] ], [ [ 23, -2 ], [ 12, -1 ] ],
[ [ 109, -20 ], [ 60, -11 ] ], [ [ 17, -4 ], [ 30, -7 ] ],
[ [ 7, -2 ], [ 18, -5 ] ], [ [ 41, -26 ], [ 30, -19 ] ],
[ [ 7, -6 ], [ 6, -5 ] ], [ [ 53, -76 ], [ 30, -43 ] ],
[ [ 31, -50 ], [ 18, -29 ] ] ]
gap> fs12_2:=FareySymbolByData([infinity,0,1,2,3,4,infinity],[1,"odd","odd","odd","odd",1]);
[ infinity, 0, 1, 2, 3, 4, infinity ]
[ 1, "odd", "odd", "odd", "odd", 1 ]
gap> GeneratorsByFareySymbol(last);
[ [ [ 1, 4 ], [ 0, 1 ] ], [ [ 1, -1 ], [ 3, -2 ] ], [ [ 4, -7 ], [ 3, -5 ] ],
[ [ 7, -19 ], [ 3, -8 ] ], [ [ 10, -37 ], [ 3, -11 ] ] ]
gap> fs12_3:=FareySymbolByData([infinity,0,1,5/4,4/3,3/2,5/3,2,7/3,5/2,8/3,11/4,14/5,3,infinity],
> [1,"even",5,4,3,3,"even","even",2,2,4,5,"even",1]);
[ infinity, 0, 1, 5/4, 4/3, 3/2, 5/3, 2, 7/3, 5/2, 8/3, 11/4, 14/5, 3,
infinity ]
[ 1, "even", 5, 4, 3, 3, "even", "even", 2, 2, 4, 5, "even", 1 ]
gap> GeneratorsByFareySymbol(last);
[ [ [ 1, 3 ], [ 0, 1 ] ], [ [ 1, -1 ], [ 2, -1 ] ],
[ [ 67, -81 ], [ 24, -29 ] ], [ [ 65, -84 ], [ 24, -31 ] ],
[ [ 19, -27 ], [ 12, -17 ] ], [ [ 17, -29 ], [ 10, -17 ] ],
[ [ 23, -53 ], [ 10, -23 ] ], [ [ 31, -75 ], [ 12, -29 ] ],
[ [ 73, -205 ], [ 26, -73 ] ] ]
gap> fs12_4:=FareySymbolByData([infinity,0,1/3,1/2,2/3,1,2,3,10/3,7/2,11/3,4,5,6,infinity],
> [1,"even",3,2,"even","even","even","even",2,3,"even","even","even",1]);
[ infinity, 0, 1/3, 1/2, 2/3, 1, 2, 3, 10/3, 7/2, 11/3, 4, 5, 6, infinity ]
[ 1, "even", 3, 2, "even", "even", "even", "even", 2, 3, "even", "even",
"even", 1 ]
gap> GeneratorsByFareySymbol(last);
[ [ [ 1, 6 ], [ 0, 1 ] ], [ [ 3, -1 ], [ 10, -3 ] ],
[ [ 43, -18 ], [ 12, -5 ] ], [ [ 41, -24 ], [ 12, -7 ] ],
[ [ 7, -5 ], [ 10, -7 ] ], [ [ 3, -5 ], [ 2, -3 ] ],
[ [ 5, -13 ], [ 2, -5 ] ], [ [ 33, -109 ], [ 10, -33 ] ],
[ [ 37, -137 ], [ 10, -37 ] ], [ [ 9, -41 ], [ 2, -9 ] ],
[ [ 11, -61 ], [ 2, -11 ] ] ]
gap> fs12_5:=FareySymbolByData([infinity,0,1/3,2/5,1/2,1,4/3,3/2,2,3,infinity],
> ["even",1,"even","even","even","even",1,"even","even","even"]);
[ infinity, 0, 1/3, 2/5, 1/2, 1, 4/3, 3/2, 2, 3, infinity ]
[ "even", 1, "even", "even", "even", "even", 1, "even", "even", "even" ]
gap> GeneratorsByFareySymbol(last);
[ [ [ 0, -1 ], [ 1, 0 ] ], [ [ 13, -3 ], [ 9, -2 ] ],
[ [ 13, -5 ], [ 34, -13 ] ], [ [ 12, -5 ], [ 29, -12 ] ],
[ [ 3, -2 ], [ 5, -3 ] ], [ [ 13, -17 ], [ 10, -13 ] ],
[ [ 8, -13 ], [ 5, -8 ] ], [ [ 5, -13 ], [ 2, -5 ] ],
[ [ 3, -10 ], [ 1, -3 ] ] ]
gap> fs12_6:=FareySymbolByData([infinity,0,1,4/3,3/2,5/3,2,3,infinity],
> [1,"even","even",2,2,"even","even",1]);
[ infinity, 0, 1, 4/3, 3/2, 5/3, 2, 3, infinity ]
[ 1, "even", "even", 2, 2, "even", "even", 1 ]
gap> GeneratorsByFareySymbol(last);
[ [ [ 1, 3 ], [ 0, 1 ] ], [ [ 1, -1 ], [ 2, -1 ] ],
[ [ 13, -17 ], [ 10, -13 ] ], [ [ 19, -27 ], [ 12, -17 ] ],
[ [ 17, -29 ], [ 10, -17 ] ], [ [ 5, -13 ], [ 2, -5 ] ] ]
gap> fs12_7:=FareySymbolByData([infinity,0,1,2,3,4,5,6,infinity],
> [1,"even","even","even","even","even","even",1]);
[ infinity, 0, 1, 2, 3, 4, 5, 6, infinity ]
[ 1, "even", "even", "even", "even", "even", "even", 1 ]
gap> GeneratorsByFareySymbol(last);
[ [ [ 1, 6 ], [ 0, 1 ] ], [ [ 1, -1 ], [ 2, -1 ] ], [ [ 3, -5 ], [ 2, -3 ] ],
[ [ 5, -13 ], [ 2, -5 ] ], [ [ 7, -25 ], [ 2, -7 ] ],
[ [ 9, -41 ], [ 2, -9 ] ], [ [ 11, -61 ], [ 2, -11 ] ] ]
gap> fs12_8:=FareySymbolByData([infinity,0,1,3/2,2,3,infinity],
> ["even","even","even","even","even","even"]);
[ infinity, 0, 1, 3/2, 2, 3, infinity ]
[ "even", "even", "even", "even", "even", "even" ]
gap> GeneratorsByFareySymbol(last);
[ [ [ 0, -1 ], [ 1, 0 ] ], [ [ 1, -1 ], [ 2, -1 ] ],
[ [ 7, -10 ], [ 5, -7 ] ], [ [ 8, -13 ], [ 5, -8 ] ],
[ [ 5, -13 ], [ 2, -5 ] ], [ [ 3, -10 ], [ 1, -3 ] ] ]
gap> fs12_9:=FareySymbolByData([infinity,0,1/4,1/3,1/2,2/3,3/4,4/5,5/6,1,infinity],
> [1,4,3,2,2,3,4,5,5,1]);
[ infinity, 0, 1/4, 1/3, 1/2, 2/3, 3/4, 4/5, 5/6, 1, infinity ]
[ 1, 4, 3, 2, 2, 3, 4, 5, 5, 1 ]
gap> GeneratorsByFareySymbol(last);
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 19, -4 ], [ 24, -5 ] ],
[ [ 17, -5 ], [ 24, -7 ] ], [ [ 7, -3 ], [ 12, -5 ] ],
[ [ 31, -25 ], [ 36, -29 ] ] ]
gap> fs12_10:=FareySymbolByData([infinity,0,1/6,1/5,1/4,2/7,1/3,2/5,1/2,4/7,7/12,3/5,2/3,5/7,3/4,4/5,5/6,1,infinity],
> [1,2,3,7,7,8,8,6,6,9,9,4,4,5,5,3,2,1]);
[ infinity, 0, 1/6, 1/5, 1/4, 2/7, 1/3, 2/5, 1/2, 4/7, 7/12, 3/5, 2/3, 5/7,
3/4, 4/5, 5/6, 1, infinity ]
[ 1, 2, 3, 7, 7, 8, 8, 6, 6, 9, 9, 4, 4, 5, 5, 3, 2, 1 ]
gap> GeneratorsByFareySymbol(last);
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 11, -1 ], [ 12, -1 ] ],
[ [ 49, -9 ], [ 60, -11 ] ], [ [ 13, -3 ], [ 48, -11 ] ],
[ [ 13, -4 ], [ 36, -11 ] ], [ [ 13, -6 ], [ 24, -11 ] ],
[ [ 85, -49 ], [ 144, -83 ] ], [ [ 25, -16 ], [ 36, -23 ] ],
[ [ 37, -27 ], [ 48, -35 ] ] ]
gap> G:=CongruenceSubgroupGamma0(20);
gap> fs:=FareySymbol(G);
[ infinity, 0, 1/5, 1/4, 2/7, 3/10, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1,
infinity ]
[ 1, 3, 4, 6, 7, 7, 5, 2, 2, 3, 6, 4, 5, 1 ]
gap> G:=PrincipalCongruenceSubgroup(2);
gap> FareySymbol(G);
[ infinity, 0, 1, 2, infinity ]
[ 2, 1, 1, 2 ]
gap> GeneratorsOfGroup(G);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 2 ], [ 0, 1 ] ], [ [ 3, -2 ], [ 2, -1 ] ] ]
gap> G:=CongruenceSubgroupGamma0(2);
gap> FareySymbol(G);
[ infinity, 0, 1, infinity ]
[ 1, "even", 1 ]
gap> GeneratorsOfGroup(G);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 1, -1 ], [ 2, -1 ] ] ]
gap> G:=CongruenceSubgroupGamma0(3);
gap> FareySymbol(G);
[ infinity, 0, 1, infinity ]
[ 1, "odd", 1 ]
gap> GeneratorsOfGroup(G);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 1, -1 ], [ 3, -2 ] ] ]
gap> G:=PrincipalCongruenceSubgroup(4);
gap> FareySymbol(G);
[ infinity, 0, 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, infinity ]
[ 1, 5, 2, 2, 3, 3, 4, 4, 5, 1 ]
gap> GeneratorsOfGroup(G);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 4 ], [ 0, 1 ] ], [ [ -15, 4 ], [ -4, 1 ] ], [ [ 5, -4 ], [ 4, -3 ] ],
[ [ 9, -16 ], [ 4, -7 ] ], [ [ 13, -36 ], [ 4, -11 ] ] ]
gap> G:=CongruenceSubgroupGamma0(4);
gap> FareySymbol(G);
[ infinity, 0, 1/2, 1, infinity ]
[ 1, 2, 2, 1 ]
gap> GeneratorsOfGroup(G);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 3, -1 ], [ 4, -1 ] ] ]
gap> G:=CongruenceSubgroupGamma0(5);
gap> FareySymbol(G);
[ infinity, 0, 1/2, 1, infinity ]
[ 1, "even", "even", 1 ]
gap> GeneratorsOfGroup(G);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 2, -1 ], [ 5, -2 ] ], [ [ 3, -2 ], [ 5, -3 ] ] ]
gap> G:=CongruenceSubgroupGamma0(6);
gap> FareySymbol(G);
[ infinity, 0, 1/3, 1/2, 2/3, 1, infinity ]
[ 1, 3, 2, 2, 3, 1 ]
gap> GeneratorsOfGroup(G);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 5, -1 ], [ 6, -1 ] ], [ [ 7, -3 ], [ 12, -5 ] ]
]
gap> G:=CongruenceSubgroupGamma0(7);
gap> FareySymbol(G);
[ infinity, 0, 1/2, 1, infinity ]
[ 1, "odd", "odd", 1 ]
gap> GeneratorsOfGroup(G);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 2, -1 ], [ 7, -3 ] ], [ [ 4, -3 ], [ 7, -5 ] ] ]
gap> G:=CongruenceSubgroupGamma0(9);
gap> FareySymbol(G);
[ infinity, 0, 1/3, 1/2, 2/3, 1, infinity ]
[ 1, 2, 2, 3, 3, 1 ]
gap> GeneratorsOfGroup(G);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 4, -1 ], [ 9, -2 ] ], [ [ 7, -4 ], [ 9, -5 ] ] ]
gap> G:=CongruenceSubgroupGamma0(10);
gap> FareySymbol(G);
[ infinity, 0, 1/3, 2/5, 1/2, 3/5, 2/3, 1, infinity ]
[ 1, "even", 3, 2, 2, 3, "even", 1 ]
gap> GeneratorsOfGroup(G);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 3, -1 ], [ 10, -3 ] ],
[ [ 19, -7 ], [ 30, -11 ] ], [ [ 11, -5 ], [ 20, -9 ] ],
[ [ 7, -5 ], [ 10, -7 ] ] ]
gap> G:=CongruenceSubgroupGamma0(13);
gap> FareySymbol(G);
[ infinity, 0, 1/3, 1/2, 2/3, 1, infinity ]
[ 1, "odd", "even", "even", "odd", 1 ]
gap> GeneratorsOfGroup(G);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 3, -1 ], [ 13, -4 ] ],
[ [ 5, -2 ], [ 13, -5 ] ], [ [ 8, -5 ], [ 13, -8 ] ],
[ [ 9, -7 ], [ 13, -10 ] ] ]
gap> G:=CongruenceSubgroupGamma0(18);
gap> FareySymbol(G);
[ infinity, 0, 1/6, 1/5, 2/9, 1/4, 1/3, 1/2, 2/3, 3/4, 7/9, 4/5, 5/6, 1,
infinity ]
[ 1, 4, 4, 7, 6, 2, 2, 3, 3, 6, 7, 5, 5, 1 ]
gap> GeneratorsOfGroup(G);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 7, -1 ], [ 36, -5 ] ],
[ [ 71, -15 ], [ 90, -19 ] ], [ [ 55, -13 ], [ 72, -17 ] ],
[ [ 7, -2 ], [ 18, -5 ] ], [ [ 13, -8 ], [ 18, -11 ] ],
[ [ 31, -25 ], [ 36, -29 ] ] ]
gap> G:=CongruenceSubgroupGamma0(25);
gap> FareySymbol(G);
[ infinity, 0, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1, infinity ]
[ 1, 2, 2, "even", 3, 3, 4, 4, "even", 5, 5, 1 ]
gap> GeneratorsOfGroup(G);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 1 ], [ 0, 1 ] ], [ [ 6, -1 ], [ 25, -4 ] ],
[ [ 7, -2 ], [ 25, -7 ] ], [ [ 11, -4 ], [ 25, -9 ] ],
[ [ 16, -9 ], [ 25, -14 ] ], [ [ 18, -13 ], [ 25, -18 ] ],
[ [ 21, -16 ], [ 25, -19 ] ] ]
gap> G:=IntersectionOfCongruenceSubgroups(PrincipalCongruenceSubgroup(2),CongruenceSubgroupGamma0(4));
gap> FareySymbol(G);
[ infinity, 0, 1/2, 1, 3/2, 2, infinity ]
[ 1, 3, 2, 2, 3, 1 ]
gap> GeneratorsOfGroup(G);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 2 ], [ 0, 1 ] ], [ [ 7, -2 ], [ 4, -1 ] ], [ [ 5, -4 ], [ 4, -3 ] ] ]
gap> G:=IntersectionOfCongruenceSubgroups(PrincipalCongruenceSubgroup(2),CongruenceSubgroupGamma0(3));
gap> FareySymbol(G);
[ infinity, 0, 1/3, 1/2, 2/3, 1, 4/3, 3/2, 5/3, 2, infinity ]
[ 1, 5, 4, 3, 2, 2, 3, 4, 5, 1 ]
gap> GeneratorsOfGroup(G);
#I Using the Congruence package for GeneratorsOfGroup ...
[ [ [ 1, 2 ], [ 0, 1 ] ], [ [ 11, -2 ], [ 6, -1 ] ],
[ [ 19, -8 ], [ 12, -5 ] ], [ [ 17, -10 ], [ 12, -7 ] ],
[ [ 7, -6 ], [ 6, -5 ] ] ]
gap> G16:=CongruenceSubgroupGamma0(16);;
gap> FS16:=FareySymbol(G16);;
gap> gens:=GeneratorsByFareySymbol(FS16);;
gap> glue_list:=__congruence_gluing_matrices(FS16);
[ 1, 2, -2, 3, -3, 4, -4, 5, -5, -1 ]
gap> for i in [1..1000] do
> g:=Random(G16);
> w:=__congruence_FactorizeMat( G16, g );
> h:=__congruence_CheckFactorizeMat(gens,w);
> if g<>h and g<>-h then
> Print("Error:", g, " is not plus/minus ", h , "\n");
> fi;
> od;
congruence-1.2.6/doc/manual.six 000644 000766 000024 00000023144 14577601110 016425 0 ustar 00obk1 staff 000000 000000 #SIXFORMAT GapDocGAP
HELPBOOKINFOSIXTMP := rec(
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entries :=
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,
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[
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,
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GAP (Congruence) - References
[CLLT93] Chan, S.-P., Lang, M.-L., Lim, C.-H. and Tan, S. P.,
Special polygons for subgroups of the modular group and
applications,
Internat. J. Math.,
4 (1)
(1993),
11--34.
[Kul91] Kulkarni, R. S.,
An arithmetic-geometric method in the study of the subgroups
of the modular group,
Amer. J. Math.,
113 (6)
(1991),
1053--1133.
[LLT95a] Lang, M.-L., Lim, C.-H. and Tan, S. P.,
An algorithm for determining if a subgroup of the modular
group is congruence,
J. London Math. Soc. (2),
51 (3)
(1995),
491--502.
[LLT95b] Lang, M.-L., Lim, C.-H. and Tan, S. P.,
Independent generators for congruence subgroups of Hecke
groups,
Math. Z.,
220 (4)
(1995),
569--594.
Congruence 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 the FSF's own site https://www.gnu.org/licenses/gpl.html.
If you obtained Congruence, we would be grateful for a short notification sent to one of the authors.
If you publish a result which was partially obtained with the usage of Congruence, please cite it in the following form:
A. Dooms, E. Jespers, O. Konovalov and H. Verrill. Congruence --- Congruence subgroups of \(SL_2(ℤ)\), Version 1.2.6; 2024 (https://gap-packages.github.io/congruence/).
Acknowledgements
We are very grateful to Mong-Lung Lang, Chong-Hai Lim and Ser Peow Tan for their comments provided while implementing algorithms from [LLT95a] and [LLT95b], and to Francqui Stichting (Belgium) for the support of the third author.
The package Congruence provides functions to construct several types of canonical congruence subgroups in SL_2(ℤ), and also intersections of a finite number of such subgroups. They will return a matrix group in the category IsCongruenceSubgroup, which is defined as a subcategory of IsMatrixGroup, and which will have a distinguishing property determining whether it is a congruence subgroup of one of the canonical types, or an intersection of such congruence subgroups (if it can not be reduced to one of the canonical congruence subgroups). To start to work with the package, you need first to load it as follows:
gap>LoadPackage("congruence");
-----------------------------------------------------------------------------
Loading Congruence 1.2.6 (Congruence subgroups of SL(2,Integers))
by Ann Dooms (http://homepages.vub.ac.be/~andooms),
Eric Jespers (http://homepages.vub.ac.be/~efjesper),
Olexandr Konovalov (https://olexandr-konovalov.github.io/), and
Helena Verrill (http://www.math.lsu.edu/~verrill).
maintained by:
Ann Dooms (http://homepages.vub.ac.be/~andooms),
Olexandr Konovalov (https://olexandr-konovalov.github.io/), and
Helena Verrill (http://www.math.lsu.edu/~verrill).
Homepage: https://gap-packages.github.io/congruence
Report issues at https://github.com/gap-packages/congruence/issues
-----------------------------------------------------------------------------
true
2.1 Construction of congruence subgroups
2.1-1 PrincipalCongruenceSubgroup
‣ PrincipalCongruenceSubgroup( N )
( operation )
Returns the principal congruence subgroup Γ(N) of level N in SL_2(ℤ).
This subgroup consists of all matrices of the form
[1+N*a N*b]
[ N*c 1+N*d]
where a,b,c,d are integers. The returned group will have the property IsPrincipalCongruenceSubgroup (2.2-1).
Returns the congruence subgroup Γ_0(N) of level N in SL_2(ℤ).
This subgroup consists of all matrices of the form
[a b]
[N*c d]
where a,b,c,d are integers. The returned group will have the property IsCongruenceSubgroupGamma0 (2.2-2).
gap>G0_4:=CongruenceSubgroupGamma0(4);
<congruence subgroup CongruenceSubgroupGamma_0(4) in SL_2(Z)>
2.1-3 CongruenceSubgroupGammaUpper0
‣ CongruenceSubgroupGammaUpper0( N )
( operation )
Returns the congruence subgroup Γ^0(N) of level N in SL_2(ℤ).
This subgroup consists of all matrices of the form
[a N*b]
[c d]
where a,b,c,d are integers. The returned group will have the property IsCongruenceSubgroupGammaUpper0 (2.2-3).
gap>GU0_2:=CongruenceSubgroupGammaUpper0(2);
<congruence subgroup CongruenceSubgroupGamma^0(2) in SL_2(Z)>
2.1-4 CongruenceSubgroupGamma1
‣ CongruenceSubgroupGamma1( N )
( operation )
Returns the congruence subgroup Γ_1(N) of level N in SL_2(ℤ).
This subgroup consists of all matrices of the form
[1+N*a b]
[ N*c 1+N*d]
where a,b,c,d are integers. The returned group will have the property IsCongruenceSubgroupGamma1 (2.2-4).
gap>G1_6:=CongruenceSubgroupGamma1(6);
<congruence subgroup CongruenceSubgroupGamma_1(6) in SL_2(Z)>
2.1-5 CongruenceSubgroupGammaUpper1
‣ CongruenceSubgroupGammaUpper1( N )
( operation )
Returns the congruence subgroup Γ^1(N) of level N in SL_2(ℤ).
This subgroup consists of all matrices of the form
[1+N*a N*b]
[ c 1+N*d]
where a,b,c,d are integers. The returned group will have the property IsCongruenceSubgroupGammaUpper1 (2.2-5).
gap>GU1_4:=CongruenceSubgroupGammaUpper1(4);
<congruence subgroup CongruenceSubgroupGamma^1(4) in SL_2(Z)>
2.1-6 IntersectionOfCongruenceSubgroups
‣ IntersectionOfCongruenceSubgroups( G1, G2, ..., GN )
( function )
‣ Intersection( G1, G2, ..., GN )
( function )
Returns the intersection of its arguments, which can be congruence subgroups or their intersections, constructed with the same function. It is not necessary for the user to use IntersectionOfCongruenceSubgroups, since it will be called automatically from Intersection.
The returned group will have the property IsIntersectionOfCongruenceSubgroups (2.2-6).
The list of congruence subgroups that form the intersection can be obtained using DefiningCongruenceSubgroups (2.3-3). Note, that when the intersection appears to be one of the canonical congruence subgroups, the package will recognize this and will return a canonical subgroup of the appropriate type.
gap>I:=IntersectionOfCongruenceSubgroups(G0_4,GU1_4);
<principal congruence subgroup of level 4 in SL_2(Z)>
gap>J:=IntersectionOfCongruenceSubgroups(G0_4,G1_6);
<intersection of congruence subgroups of resulting level 12 in SL_2(Z)>
2.2 Properties of congruence subgroups
A congruence subgroup constructed by one of the five above listed functions will have certain properties determining its type. These properties will be used for method selection by Congruence algorithms. Note that they do not provide an actual test whether a certain matrix group is a congruence subgroup or not.
2.2-1 IsPrincipalCongruenceSubgroup
‣ IsPrincipalCongruenceSubgroup( G )
( property )
For a congruence subgroup G in the category IsCongruenceSubgroup, returns true if G was constructed by PrincipalCongruenceSubgroup (2.1-1) (or reduced to one as a result of an intersection) and returns false otherwise.
For a congruence subgroup G in the category IsCongruenceSubgroup, returns true if G was constructed by CongruenceSubgroupGamma0 (2.1-2) (or reduced to one as a result of an intersection) and returns false otherwise.
2.2-3 IsCongruenceSubgroupGammaUpper0
‣ IsCongruenceSubgroupGammaUpper0( G )
( property )
For a congruence subgroup G in the category IsCongruenceSubgroup, returns true if G was constructed by CongruenceSubgroupGammaUpper0 (2.1-3) (or reduced to one as a result of an intersection) and returns false otherwise.
2.2-4 IsCongruenceSubgroupGamma1
‣ IsCongruenceSubgroupGamma1( G )
( property )
For a congruence subgroup G in the category IsCongruenceSubgroup, returns true if G was constructed by CongruenceSubgroupGamma1 (2.1-4) (or reduced to one as a result of an intersection) and returns false otherwise.
2.2-5 IsCongruenceSubgroupGammaUpper1
‣ IsCongruenceSubgroupGammaUpper1( G )
( property )
For a congruence subgroup G in the category IsCongruenceSubgroup, returns true if G was constructed by CongruenceSubgroupGammaUpper1 (2.1-5) (or reduced to one as a result of an intersection) and returns false otherwise.
2.2-6 IsIntersectionOfCongruenceSubgroups
‣ IsIntersectionOfCongruenceSubgroups( G )
( property )
For a congruence subgroup G in the category IsCongruenceSubgroup, returns true if G was constructed by IntersectionOfCongruenceSubgroups (2.1-6) and without being one of the canonical congruence subgroups, otherwise it returns false.
The next three attributes store key properties of congruence subgroups.
2.3-1 LevelOfCongruenceSubgroup
‣ LevelOfCongruenceSubgroup( G )
( attribute )
Stores the level of the congruence subgroup G. The (arithmetic) level of a congruence subgroup G is the smallest positive number N such that G contains the principal congruence subgroup of level N.
Stores the index of the congruence subgroup G in SL_2(ℤ).
gap>IndexInSL2Z(G_8);
384
gap>G_2:=PrincipalCongruenceSubgroup(2);
<principal congruence subgroup of level 2 in SL_2(Z)>
gap>IndexInSL2Z(G_2);
12
gap>IndexInSL2Z(GU1_4);
12
2.3-3 DefiningCongruenceSubgroups
‣ DefiningCongruenceSubgroups( G )
( attribute )
Returns: list of congruence subgroups
For an intersection of congruence subgroups, returns the list of congruence subgroups forming this intersection. For a canonical congruence subgroup returns a list of length one containing that subgroup.
gap>DefiningCongruenceSubgroups(J);
[ <congruence subgroup CongruenceSubgroupGamma_0(4) in SL_2(Z)>,
<congruence subgroup CongruenceSubgroupGamma_1(6) in SL_2(Z)> ]
gap>P:=PrincipalCongruenceSubgroup(6);
<principal congruence subgroup of level 6 in SL_2(Z)>
gap>Q:=PrincipalCongruenceSubgroup(10);
<principal congruence subgroup of level 10 in SL_2(Z)>
gap>G:=IntersectionOfCongruenceSubgroups(Q,P);
<principal congruence subgroup of level 30 in SL_2(Z)>
gap>DefiningCongruenceSubgroups(G);
[ <principal congruence subgroup of level 30 in SL_2(Z)> ]
2.4 Operations for congruence subgroups
Congruence installs several special methods for operations already available in GAP.
2.4-1 Random
‣ Random( G )
( operation )
‣ Random( G, m )
( operation )
For a congruence subgroup G in the category IsCongruenceSubgroup, returns random element. In the two-argument form, the second parameter will control the absolute value of randomly selected entries of the matrix.
gap>Random(G_2) in G_2;
true
gap>Random(G_8,2) in G_8;
true
2.4-2 \in
‣ \in( m, G )
( operation )
It is easy to implement the membership test for congruence subgroups and their intersections.
For congruence subgroups G,H in the category IsCongruenceSubgroup, returns true if G and H are of the same type listed in PrincipalCongruenceSubgroup (2.1-1) --> CongruenceSubgroupGammaUpper1 (2.1-5) and have the same LevelOfCongruenceSubgroup (2.3-1) or if G and H are of the type IntersectionOfCongruenceSubgroups (2.1-6) and the groups from DefiningCongruenceSubgroups (2.3-3) are in one to one correspondence, otherwise it returns false.
Congruence provides methods for IsSubset for congruence subgroups. IsSubset returns true if H is a subset of G. These methods make it possible to use IsSubgroup operation for congruence subgroups.
If a congruence subgroup H is a subgroup of a congruence subgroup G, we can easily compute the index of H in G, since we know the index of both subgroups in SL_2(ℤ).
A Farey symbol is a compact and useful way to represent a subgroup of finite index in SL_2(ℤ) from which one can deduce independent generators for this subgroup. It consists of two components, namely a so-called generalised Farey sequence (gfs) and an ordered list of labels, giving additional structure to the gfs.
A generalised Farey sequence (g.F.S.) is an ordered list of the form -infinity, x_0, x_1, ... , x_n, infinity, where
1. the x_i = a_i/b_i are rational numbers in reduced form arranged in increasing order for i = 0, ... , n;
2. x_0, ... , x_n ∈ Z, and some x_i = 0;
3. we define x_-1=-infinity=-1/0 and x_n+1=infinity=1/0;
4. a_i+1b_i-a_ib_i+1=1 for i=-1, ... ,n.
The ordered list of labels of a Farey symbol gives an additional structure to the gfs. The labels correspond to each consecutive pair of x_i's and are of the following types:
1. even,
2. odd,
3. a natural number, which occurs in the list of labels exactly twice or not at all.
Note that the actual values of numerical labels are not important; it is the pairing of two intervals that matters.
The package Congruence provides functions to construct Farey symbols by the given generalised Farey sequence and corresponding list of labels. The returned Farey symbol will belong to the category IsFareySymbol and will have the representation IsFareySymbolDefaultRep.
3.1 Construction of Farey symbols
3.1-1 FareySymbolByData
‣ FareySymbolByData( gfs, labels )
( function )
This constructor creates the Farey symbol with the given generalized Farey sequence and list of labels. It also checks conditions from the definition of Farey symbol and returns an error if they are not satisfied. The data used to create the Farey symbol are stored as its attributes GeneralizedFareySequence (3.2-1) and LabelsOfFareySymbol (3.2-4).
Returns the numerator of the i-th term of the generalised Farey sequence gfs: for the 1st infinite entry returns -1, for the last one returns 1, for all other entries returns the usual numerator.
Returns the denominator of the i-th term of the generalised Farey sequence gfs: for both infinite entries returns 0, for the other ones returns the usual denominator.
The GAP package Congruence provides functions to construct several types of canonical congruence subgroups in \(SL_2(ℤ)\), and also intersections of a finite number of such subgroups.
Furthermore, it implements the algorithm for generating Farey symbols for congruence subgroups and using them to produce a system of independent generators for these subgroups.
Using the package, one can also determine indices of congruence subgroups and their intersections in \(SL_2(ℤ)\) and in other congruence subgroups, generate their random elements and check element memberships. Success of other group theoretical constructions mostly depends on whether they could be expressed in terms of group generators or not.
Congruence does not use external binaries and, therefore, works without restrictions on the operating system. It requires at least version GAPÂ 4.5, and no compatibility with previous releases of GAPÂ 4 is guaranteed.
Installation of the package is standard and follows the guidelines from the GAP manual (see Reference: Installing a GAP Package. After the package is installed, you can start GAP and load the Congruence package using the command:
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congruence-1.2.6/doc/chapInd.html 000644 000766 000024 00000014173 14577601110 016661 0 ustar 00obk1 staff 000000 000000
GAP (Congruence) - Index
A Farey symbol is a compact and useful way to represent a subgroup of finite index in \(SL_2(ℤ)\) from which one can deduce independent generators for this subgroup. It consists of two components, namely a so-called generalised Farey sequence (gfs) and an ordered list of labels, giving additional structure to the gfs.
A generalised Farey sequence (g.F.S.) is an ordered list of the form \({ -infinity, x_0, x_1, ... , x_n, infinity }\), where
1. the \(x_i = a_i/b_i\) are rational numbers in reduced form arranged in increasing order for \(i = 0, ... , n\);
2. \(x_0, ... , x_n \in Z\), and some \(x_i = 0\);
3. we define \(x_{-1}=-infinity=-1/0\) and \(x_{n+1}=infinity=1/0\);
4. \(a_{i+1}b_{i}-a_{i}b_{i+1}=1\) for \(i=-1, ... ,n\).
The ordered list of labels of a Farey symbol gives an additional structure to the gfs. The labels correspond to each consecutive pair of \(x_i\)'s and are of the following types:
1. even,
2. odd,
3. a natural number, which occurs in the list of labels exactly twice or not at all.
Note that the actual values of numerical labels are not important; it is the pairing of two intervals that matters.
The package Congruence provides functions to construct Farey symbols by the given generalised Farey sequence and corresponding list of labels. The returned Farey symbol will belong to the category IsFareySymbol and will have the representation IsFareySymbolDefaultRep.
3.1 Construction of Farey symbols
3.1-1 FareySymbolByData
‣ FareySymbolByData( gfs, labels )
( function )
This constructor creates the Farey symbol with the given generalized Farey sequence and list of labels. It also checks conditions from the definition of Farey symbol and returns an error if they are not satisfied. The data used to create the Farey symbol are stored as its attributes GeneralizedFareySequence (3.2-1) and LabelsOfFareySymbol (3.2-4).
Returns the numerator of the i-th term of the generalised Farey sequence gfs: for the 1st infinite entry returns -1, for the last one returns 1, for all other entries returns the usual numerator.
Returns the denominator of the i-th term of the generalised Farey sequence gfs: for both infinite entries returns 0, for the other ones returns the usual denominator.
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-------------------------------------------------------
congruence-1.2.6/doc/service.xml 000644 000766 000024 00000001426 14577601060 016610 0 ustar 00obk1 staff 000000 000000 Service functions of the &Congruence; packageAdditional information displayed by &Congruence; algorithmsInfoCongruence is a special Info class for &Congruence; algorithms.
It has 3 levels: 0, 1 (default) and 2. To change the info level to k,
use the command SetInfoLevel(InfoCongruence, k).
In the example below we use this mechanism to see more details during
the Farey symbol construction for a congruence subgroup.
congruence-1.2.6/doc/cong.xml 000644 000766 000024 00000041126 14577601060 016077 0 ustar 00obk1 staff 000000 000000 Construction of congruence subgroupsIsCongruenceSubgroup
The package &Congruence; provides functions to construct several
types of canonical congruence subgroups in SL_2(&ZZ;), and also
intersections of a finite number of such subgroups. They will return
a matrix group in the category IsCongruenceSubgroup,
which is defined as a subcategory of IsMatrixGroup, and
which will have a distinguishing property determining whether it is
a congruence subgroup of one of the canonical types, or an intersection
of such congruence subgroups (if it can not be reduced to one of the
canonical congruence subgroups).
To start to work with the package, you need first to load it as follows:
LoadPackage("congruence");
-----------------------------------------------------------------------------
Loading Congruence 1.2.6 (Congruence subgroups of SL(2,Integers))
by Ann Dooms (http://homepages.vub.ac.be/~andooms),
Eric Jespers (http://homepages.vub.ac.be/~efjesper),
Olexandr Konovalov (https://olexandr-konovalov.github.io/), and
Helena Verrill (http://www.math.lsu.edu/~verrill).
maintained by:
Ann Dooms (http://homepages.vub.ac.be/~andooms),
Olexandr Konovalov (https://olexandr-konovalov.github.io/), and
Helena Verrill (http://www.math.lsu.edu/~verrill).
Homepage: https://gap-packages.github.io/congruence
Report issues at https://github.com/gap-packages/congruence/issues
-----------------------------------------------------------------------------
true
]]>
Construction of congruence subgroups
Returns the principal congruence subgroup \Gamma(N) of level N
in SL_2(&ZZ;).
This subgroup consists of
all matrices of the form
where a,b,c,d are integers.
The returned group will have the property
.
G_8:=PrincipalCongruenceSubgroup(8);
gap> IsGroup(G_8);
true
gap> IsMatrixGroup(G_8);
true
gap> DimensionOfMatrixGroup(G_8);
2
gap> MultiplicativeNeutralElement(G_8);
[ [ 1, 0 ], [ 0, 1 ] ]
gap> One(G);
[ [ 1, 0 ], [ 0, 1 ] ]
gap> [[1,2],[3,4]] in G_8;
false
gap> [[1,8],[8,65]] in G_8;
true
gap> SL_2:=SL(2,Integers);
SL(2,Integers)
gap> IsSubgroup(SL_2,G_8);
true
]]>
Returns the congruence subgroup \Gamma_0(N)
of level N in SL_2(&ZZ;).
This subgroup consists of
all matrices of the form
where a,b,c,d are integers.
The returned group will have the property
.
G0_4:=CongruenceSubgroupGamma0(4);
]]>
Returns the congruence subgroup \Gamma^0(N)
of level N in SL_2(&ZZ;).
This subgroup consists of
all matrices of the form
where a,b,c,d are integers.
The returned group will have the property
.
GU0_2:=CongruenceSubgroupGammaUpper0(2);
]]>
Returns the congruence subgroup \Gamma_1(N)
of level N in SL_2(&ZZ;).
This subgroup consists of
all matrices of the form
where a,b,c,d are integers.
The returned group will have the property
.
G1_6:=CongruenceSubgroupGamma1(6);
]]>
Returns the congruence subgroup \Gamma^1(N)
of level N in SL_2(&ZZ;).
This subgroup consists of
all matrices of the form
where a,b,c,d are integers.
The returned group will have the property
.
GU1_4:=CongruenceSubgroupGammaUpper1(4);
]]>
Returns the intersection of its arguments, which can be
congruence subgroups or their intersections, constructed
with the same function. It is not necessary for the user
to use IntersectionOfCongruenceSubgroups, since
it will be called automatically from Intersection.
The returned group will have the property
.
The list of congruence subgroups that form the intersection
can be obtained using .
Note, that when the intersection appears to be one of the
canonical congruence subgroups, the package will recognize
this and will return a canonical subgroup of the
appropriate type.
I:=IntersectionOfCongruenceSubgroups(G0_4,GU1_4);
gap> J:=IntersectionOfCongruenceSubgroups(G0_4,G1_6);
]]>
Properties of congruence subgroups
A congruence subgroup constructed by one of the five above listed functions
will have certain properties determining its type. These properties will be
used for method selection by &Congruence; algorithms. Note that they do not
provide an actual test whether a certain matrix group is a congruence subgroup
or not.
For a congruence subgroup G in the category
IsCongruenceSubgroup, returns true if G was
constructed by
(or reduced to one as a result of an intersection) and
returns false otherwise.
IsPrincipalCongruenceSubgroup(G_8);
true
gap> IsPrincipalCongruenceSubgroup(G0_4);
false
gap> IsPrincipalCongruenceSubgroup(I);
true
]]>
For a congruence subgroup G in the category
IsCongruenceSubgroup, returns true if
G was constructed by (or reduced to one as a result of an intersection)
and returns false otherwise.
For a congruence subgroup G in the category
IsCongruenceSubgroup, returns true if
G was constructed by (or reduced to one as a result of an intersection)
and returns false otherwise.
For a congruence subgroup G in the category
IsCongruenceSubgroup, returns true if
G was constructed by (or reduced to one as a result of an intersection)
and returns false otherwise.
For a congruence subgroup G in the category
IsCongruenceSubgroup, returns true if
G was constructed by (or reduced to one as a result of an intersection)
and returns false otherwise.
For a congruence subgroup G in the category
IsCongruenceSubgroup, returns true if
G was constructed by
and
without being one of the canonical congruence
subgroups, otherwise it returns false.
IsIntersectionOfCongruenceSubgroups(I);
false
gap> IsIntersectionOfCongruenceSubgroups(J);
true
]]>
Attributes of congruence subgroups
The next three attributes store key properties of congruence subgroups.
Stores the level of the congruence subgroup G. The (arithmetic) level of a congruence subgroup G is the smallest positive
number N such that G contains the principal congruence subgroup of level N.
LevelOfCongruenceSubgroup(G_8);
8
gap> LevelOfCongruenceSubgroup(G1_6);
6
gap> LevelOfCongruenceSubgroup(I);
4
gap> LevelOfCongruenceSubgroup(J);
12
]]>
Stores the index of the congruence subgroup G in SL_2(&ZZ;).
IndexInSL2Z(G_8);
384
gap> G_2:=PrincipalCongruenceSubgroup(2);
gap> IndexInSL2Z(G_2);
12
gap> IndexInSL2Z(GU1_4);
12
]]>
list of congruence subgroups
For an intersection of congruence subgroups, returns
the list of congruence subgroups forming this intersection.
For a canonical congruence subgroup returns a list of length
one containing that subgroup.
DefiningCongruenceSubgroups(J);
[ ,
]
gap> P:=PrincipalCongruenceSubgroup(6);
gap> Q:=PrincipalCongruenceSubgroup(10);
gap> G:=IntersectionOfCongruenceSubgroups(Q,P);
gap> DefiningCongruenceSubgroups(G);
[ ]
]]>
Operations for congruence subgroups
&Congruence; installs several special methods for operations already
available in &GAP;.
For a congruence subgroup G in the category
IsCongruenceSubgroup, returns random element.
In the two-argument form, the second parameter will
control the absolute value of randomly selected
entries of the matrix.
Random(G_2) in G_2;
true
gap> Random(G_8,2) in G_8;
true
]]>
It is easy to implement the membership test for
congruence subgroups and their intersections.
\in([ [ 21, 10 ], [ 2, 1 ] ],G_2);
true
gap> \in([ [ 21, 10 ], [ 2, 1 ] ],G_8);
false
]]>
For congruence subgroups G,H in the category
IsCongruenceSubgroup, returns true if
G and H are of the same type listed in
-->
and have the same
or if G and H
are of the type and
the groups from
are in one
to one correspondence, otherwise it returns false.
CanEasilyCompareCongruenceSubgroups(G_8,I);
false
]]>
&Congruence; provides methods for IsSubset
for congruence subgroups. IsSubset returns true
if H is a subset of G. These methods make it possible
to use IsSubgroup operation for congruence subgroups.
IsSubset(G_2,G_8);
true
gap> IsSubset(G_8,G_2);
false
gap> f:=[PrincipalCongruenceSubgroup,CongruenceSubgroupGamma1,CongruenceSubgroupGammaUpper1,CongruenceSubgroupGamma0,CongruenceSubgroupGammaUpper0];;
gap> g1:=List(f, t -> t(2));;
gap> g2:=List(f, t -> t(4));;
gap> for g in g2 do
> Print( List( g1, x -> IsSubgroup(x,g) ), "\n");
> od;
[ true, true, true, true, true ]
[ false, true, false, true, false ]
[ false, false, true, false, true ]
[ false, false, false, true, false ]
[ false, false, false, false, true ]
]]>
If a congruence subgroup H is a subgroup of a congruence
subgroup G, we can easily compute the index of H in
G, since we know the index of both subgroups in SL_2(&ZZ;).
Index(G_2,G_8);
32
]]>
congruence-1.2.6/doc/chap4.html 000644 000766 000024 00000037101 14577601110 016306 0 ustar 00obk1 staff 000000 000000
GAP (Congruence) - Chapter 4: Farey symbols for congruence subgroups
The package Congruence provides functions to construct Farey symbols for finite index subgroups. The algorithm used in the package allows to construct a Farey symbol for any finite index subgroup of SL_2(ℤ) for which it is possible to check whether a given matrix belongs to this subgroup or not.
The development of an algorithm to determine the Farey symbol for a subgroup G of a finite index in SL_2(ℤ) was started by Ravi Kulkarni in [Kul91] and later it was improved by Mong-Lung Lang, Chong-Hai Lim and Ser-Peow Tan in [LLT95b], [LLT95a].
4.1 Computation of the Farey symbol for a finite index subgroup
4.1-1 FareySymbol
‣ FareySymbol( G )
( attribute )
For a subgroup of a finite index G, this attribute stores one of the Farey symbols corresponding to the congruence subgroup G. The algorithm for its computation will work for any matrix group for which a membership test is available.
4.2 Computation of generators of a finite index subgroup from its Farey symbol
If fs is the Farey symbol for a group G with r_1 even labels, r_2 odd labels and r_3 pairs of intervals, then G is generated by r_1+r_2+r_3 matrices, which form a set of independent generators for G. These matrices are constructed as follows:
for each even interval [x_i, x_i+1], take the matrix
By Proposition 7.2 in [Kulkarni], for the Farey symbol with underlying generalized Farey sequence [infinity, x0, x1, ..., xn, infinity], the index in PSL_2(Z) is given by the formula d = 3*n + e3, where e3 is the number of odd intervals.
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congruence-1.2.6/doc/intro.xml 000644 000766 000024 00000004032 14577601060 016277 0 ustar 00obk1 staff 000000 000000 IntroductionGeneral aims of &Congruence; packageIsCongruenceSubgroup
The &GAP; package &Congruence; provides functions to construct several
types of canonical congruence subgroups in SL_2(&ZZ;), and also
intersections of a finite number of such subgroups.
Furthermore, it implements the algorithm for generating Farey symbols for
congruence subgroups and using them to produce a system of independent
generators for these subgroups.
Using the package, one can also determine indices of congruence subgroups and
their intersections in SL_2(&ZZ;) and in other congruence subgroups,
generate their random elements and check element memberships. Success of other
group theoretical constructions mostly depends on whether they could be
expressed in terms of group generators or not.
For the theoretical backround, we refer to ,
, and .
Installation and system requirements
&Congruence; is distributed in standard formats (tar.gz,
tar.bz2, -win.zip) and can be obtained from
https://gap-packages.github.io/congruence/.
&Congruence; does not use external binaries and, therefore, works without
restrictions on the operating system.
It requires at least version &GAP; 4.5, and no compatibility with
previous releases of &GAP; 4 is guaranteed.
Installation of the package is standard and follows the guidelines from the
&GAP; manual (see .
After the package is installed, you can start &GAP; and load
the &Congruence; package using the command:
LoadPackage("congruence");
]]>
congruence-1.2.6/doc/chap2_mj.html 000644 000766 000024 00000101365 14577601110 016776 0 ustar 00obk1 staff 000000 000000
GAP (Congruence) - Chapter 2: Construction of congruence subgroups
The package Congruence provides functions to construct several types of canonical congruence subgroups in \(SL_2(ℤ)\), and also intersections of a finite number of such subgroups. They will return a matrix group in the category IsCongruenceSubgroup, which is defined as a subcategory of IsMatrixGroup, and which will have a distinguishing property determining whether it is a congruence subgroup of one of the canonical types, or an intersection of such congruence subgroups (if it can not be reduced to one of the canonical congruence subgroups). To start to work with the package, you need first to load it as follows:
gap>LoadPackage("congruence");
-----------------------------------------------------------------------------
Loading Congruence 1.2.6 (Congruence subgroups of SL(2,Integers))
by Ann Dooms (http://homepages.vub.ac.be/~andooms),
Eric Jespers (http://homepages.vub.ac.be/~efjesper),
Olexandr Konovalov (https://olexandr-konovalov.github.io/), and
Helena Verrill (http://www.math.lsu.edu/~verrill).
maintained by:
Ann Dooms (http://homepages.vub.ac.be/~andooms),
Olexandr Konovalov (https://olexandr-konovalov.github.io/), and
Helena Verrill (http://www.math.lsu.edu/~verrill).
Homepage: https://gap-packages.github.io/congruence
Report issues at https://github.com/gap-packages/congruence/issues
-----------------------------------------------------------------------------
true
2.1 Construction of congruence subgroups
2.1-1 PrincipalCongruenceSubgroup
‣ PrincipalCongruenceSubgroup( N )
( operation )
Returns the principal congruence subgroup \(\Gamma(N)\) of level N in \(SL_2(ℤ)\).
This subgroup consists of all matrices of the form
[1+N*a N*b]
[ N*c 1+N*d]
where \(a\),\(b\),\(c\),\(d\) are integers. The returned group will have the property IsPrincipalCongruenceSubgroup (2.2-1).
Returns the congruence subgroup \(\Gamma_0(N)\) of level N in \(SL_2(ℤ)\).
This subgroup consists of all matrices of the form
[a b]
[N*c d]
where \(a\),\(b\),\(c\),\(d\) are integers. The returned group will have the property IsCongruenceSubgroupGamma0 (2.2-2).
gap>G0_4:=CongruenceSubgroupGamma0(4);
<congruence subgroup CongruenceSubgroupGamma_0(4) in SL_2(Z)>
2.1-3 CongruenceSubgroupGammaUpper0
‣ CongruenceSubgroupGammaUpper0( N )
( operation )
Returns the congruence subgroup \(\Gamma^0(N)\) of level N in \(SL_2(ℤ)\).
This subgroup consists of all matrices of the form
[a N*b]
[c d]
where \(a\),\(b\),\(c\),\(d\) are integers. The returned group will have the property IsCongruenceSubgroupGammaUpper0 (2.2-3).
gap>GU0_2:=CongruenceSubgroupGammaUpper0(2);
<congruence subgroup CongruenceSubgroupGamma^0(2) in SL_2(Z)>
2.1-4 CongruenceSubgroupGamma1
‣ CongruenceSubgroupGamma1( N )
( operation )
Returns the congruence subgroup \(\Gamma_1(N)\) of level N in \(SL_2(ℤ)\).
This subgroup consists of all matrices of the form
[1+N*a b]
[ N*c 1+N*d]
where \(a\),\(b\),\(c\),\(d\) are integers. The returned group will have the property IsCongruenceSubgroupGamma1 (2.2-4).
gap>G1_6:=CongruenceSubgroupGamma1(6);
<congruence subgroup CongruenceSubgroupGamma_1(6) in SL_2(Z)>
2.1-5 CongruenceSubgroupGammaUpper1
‣ CongruenceSubgroupGammaUpper1( N )
( operation )
Returns the congruence subgroup \(\Gamma^1(N)\) of level N in \(SL_2(ℤ)\).
This subgroup consists of all matrices of the form
[1+N*a N*b]
[ c 1+N*d]
where \(a\),\(b\),\(c\),\(d\) are integers. The returned group will have the property IsCongruenceSubgroupGammaUpper1 (2.2-5).
gap>GU1_4:=CongruenceSubgroupGammaUpper1(4);
<congruence subgroup CongruenceSubgroupGamma^1(4) in SL_2(Z)>
2.1-6 IntersectionOfCongruenceSubgroups
‣ IntersectionOfCongruenceSubgroups( G1, G2, ..., GN )
( function )
‣ Intersection( G1, G2, ..., GN )
( function )
Returns the intersection of its arguments, which can be congruence subgroups or their intersections, constructed with the same function. It is not necessary for the user to use IntersectionOfCongruenceSubgroups, since it will be called automatically from Intersection.
The returned group will have the property IsIntersectionOfCongruenceSubgroups (2.2-6).
The list of congruence subgroups that form the intersection can be obtained using DefiningCongruenceSubgroups (2.3-3). Note, that when the intersection appears to be one of the canonical congruence subgroups, the package will recognize this and will return a canonical subgroup of the appropriate type.
gap>I:=IntersectionOfCongruenceSubgroups(G0_4,GU1_4);
<principal congruence subgroup of level 4 in SL_2(Z)>
gap>J:=IntersectionOfCongruenceSubgroups(G0_4,G1_6);
<intersection of congruence subgroups of resulting level 12 in SL_2(Z)>
2.2 Properties of congruence subgroups
A congruence subgroup constructed by one of the five above listed functions will have certain properties determining its type. These properties will be used for method selection by Congruence algorithms. Note that they do not provide an actual test whether a certain matrix group is a congruence subgroup or not.
2.2-1 IsPrincipalCongruenceSubgroup
‣ IsPrincipalCongruenceSubgroup( G )
( property )
For a congruence subgroup G in the category IsCongruenceSubgroup, returns true if G was constructed by PrincipalCongruenceSubgroup (2.1-1) (or reduced to one as a result of an intersection) and returns false otherwise.
For a congruence subgroup G in the category IsCongruenceSubgroup, returns true if G was constructed by CongruenceSubgroupGamma0 (2.1-2) (or reduced to one as a result of an intersection) and returns false otherwise.
2.2-3 IsCongruenceSubgroupGammaUpper0
‣ IsCongruenceSubgroupGammaUpper0( G )
( property )
For a congruence subgroup G in the category IsCongruenceSubgroup, returns true if G was constructed by CongruenceSubgroupGammaUpper0 (2.1-3) (or reduced to one as a result of an intersection) and returns false otherwise.
2.2-4 IsCongruenceSubgroupGamma1
‣ IsCongruenceSubgroupGamma1( G )
( property )
For a congruence subgroup G in the category IsCongruenceSubgroup, returns true if G was constructed by CongruenceSubgroupGamma1 (2.1-4) (or reduced to one as a result of an intersection) and returns false otherwise.
2.2-5 IsCongruenceSubgroupGammaUpper1
‣ IsCongruenceSubgroupGammaUpper1( G )
( property )
For a congruence subgroup G in the category IsCongruenceSubgroup, returns true if G was constructed by CongruenceSubgroupGammaUpper1 (2.1-5) (or reduced to one as a result of an intersection) and returns false otherwise.
2.2-6 IsIntersectionOfCongruenceSubgroups
‣ IsIntersectionOfCongruenceSubgroups( G )
( property )
For a congruence subgroup G in the category IsCongruenceSubgroup, returns true if G was constructed by IntersectionOfCongruenceSubgroups (2.1-6) and without being one of the canonical congruence subgroups, otherwise it returns false.
The next three attributes store key properties of congruence subgroups.
2.3-1 LevelOfCongruenceSubgroup
‣ LevelOfCongruenceSubgroup( G )
( attribute )
Stores the level of the congruence subgroup G. The (arithmetic) level of a congruence subgroup G is the smallest positive number N such that G contains the principal congruence subgroup of level N.
Stores the index of the congruence subgroup G in \(SL_2(ℤ)\).
gap>IndexInSL2Z(G_8);
384
gap>G_2:=PrincipalCongruenceSubgroup(2);
<principal congruence subgroup of level 2 in SL_2(Z)>
gap>IndexInSL2Z(G_2);
12
gap>IndexInSL2Z(GU1_4);
12
2.3-3 DefiningCongruenceSubgroups
‣ DefiningCongruenceSubgroups( G )
( attribute )
Returns: list of congruence subgroups
For an intersection of congruence subgroups, returns the list of congruence subgroups forming this intersection. For a canonical congruence subgroup returns a list of length one containing that subgroup.
gap>DefiningCongruenceSubgroups(J);
[ <congruence subgroup CongruenceSubgroupGamma_0(4) in SL_2(Z)>,
<congruence subgroup CongruenceSubgroupGamma_1(6) in SL_2(Z)> ]
gap>P:=PrincipalCongruenceSubgroup(6);
<principal congruence subgroup of level 6 in SL_2(Z)>
gap>Q:=PrincipalCongruenceSubgroup(10);
<principal congruence subgroup of level 10 in SL_2(Z)>
gap>G:=IntersectionOfCongruenceSubgroups(Q,P);
<principal congruence subgroup of level 30 in SL_2(Z)>
gap>DefiningCongruenceSubgroups(G);
[ <principal congruence subgroup of level 30 in SL_2(Z)> ]
2.4 Operations for congruence subgroups
Congruence installs several special methods for operations already available in GAP.
2.4-1 Random
‣ Random( G )
( operation )
‣ Random( G, m )
( operation )
For a congruence subgroup G in the category IsCongruenceSubgroup, returns random element. In the two-argument form, the second parameter will control the absolute value of randomly selected entries of the matrix.
gap>Random(G_2) in G_2;
true
gap>Random(G_8,2) in G_8;
true
2.4-2 \in
‣ \in( m, G )
( operation )
It is easy to implement the membership test for congruence subgroups and their intersections.
For congruence subgroups G,H in the category IsCongruenceSubgroup, returns true if G and H are of the same type listed in PrincipalCongruenceSubgroup (2.1-1) --> CongruenceSubgroupGammaUpper1 (2.1-5) and have the same LevelOfCongruenceSubgroup (2.3-1) or if G and H are of the type IntersectionOfCongruenceSubgroups (2.1-6) and the groups from DefiningCongruenceSubgroups (2.3-3) are in one to one correspondence, otherwise it returns false.
Congruence provides methods for IsSubset for congruence subgroups. IsSubset returns true if H is a subset of G. These methods make it possible to use IsSubgroup operation for congruence subgroups.
If a congruence subgroup H is a subgroup of a congruence subgroup G, we can easily compute the index of H in G, since we know the index of both subgroups in \(SL_2(ℤ)\).
5.1 Additional information displayed by Congruence algorithms
5.1-1 InfoCongruence
‣ InfoCongruence
( info class )
InfoCongruence is a special Info class for Congruence algorithms. It has 3 levels: 0, 1 (default) and 2. To change the info level to k, use the command SetInfoLevel(InfoCongruence, k).
In the example below we use this mechanism to see more details during the Farey symbol construction for a congruence subgroup.