[BMMW00] Birget, J. -.C., Margolis, S., Meakin, J. and Weil, P., PSPACE-complete problems for subgroups of free groups and inverse finite automata, Theoretical Computer Science, 242 (2000), 247--281.
[LS77] Lyndon, R. C. and Schupp, P. E., Combinatorial Group Theory, Springer (1977).
[Neu33] Neumann, B., Die Automorphismengruppe der freien Gruppen, Math. Annalen, 107 (1933), 367--386.
[Nie24] Nielsen, J., Die Isomorphismengruppe der freien Gruppen, Math. Annalen, 91 (1924), 169--209.
[Sie03] Sievers, C., Algorithmen f\accent127ur freie Gruppen, Diplomarbeit, TU Braunschweig (2003).
[Sim94] Sims, C. C., Computation with Finitely Presented Groups, Cambridge University Press, Encyclopedia of Mathematics and its Applications, 48, Cambridge (1994), xiii+604 pages.
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4 April 2023
Note: This version of FGA is a fork of the original FGA, maintained by the GAP Team.
Christian Sievers
Email: c.sievers@tu-bs.de
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fga-1.5.0/doc/chap2.html 000644 000766 000024 00000067150 14413016355 015204 0 ustar 00mhorn staff 000000 000000This chapter describes methods available from the FGA package.
In the following, let f be a free group created by FreeGroup(n), and let u, u1 and u2 be finitely generated subgroups of f created by Group or Subgroup, or computed from some other subgroup of f. Let elm be an element of f.
For example:
gap> f := FreeGroup( 2 ); <free group on the generators [ f1, f2 ]> gap> u := Group( f.1^2, f.2^2, f.1*f.2 ); Group([ f1^2, f2^2, f1*f2 ]) gap> u1 := Subgroup( u, [f.1^2, f.1^4*f.2^6] ); Group([ f1^2, f1^4*f2^6 ]) gap> elm := f.1; f1 gap> u2 := Normalizer( u, elm ); Group([ f1^2 ])
These new operations are available for finitely generated subgroups of free groups:
‣ FreeGeneratorsOfGroup( u ) | ( attribute ) |
returns a list of free generators of the finitely generated subgroup u of a free group.
The elements in this list form an N-reduced set. In addition to being a free (and thus minimal) generating set for u, this means that whenever v1, v2 and v3 are elements or inverses of elements of this list, then
v1v2≠1 implies |v1v2|≥max(|v1|, |v2|), and
v1v2≠1 and v2v3≠1 implies |v1v2v3| > |v1| - |v2| + |v3|
hold, where |.| denotes the word length.
‣ RankOfFreeGroup( u ) | ( attribute ) |
‣ Rank( u ) | ( operation ) |
returns the rank of the finitely generated subgroup u of a free group.
‣ CyclicallyReducedWord( elm ) | ( operation ) |
returns the cyclically reduced form of elm.
This section lists operations that are already known to GAP. FGA installs new methods for them so that they can also be used with free groups. In cases where FGA installs methods that are usually only used internally, user functions are shown instead.
‣ Normalizer( u1, u2 ) | ( operation ) |
‣ Normalizer( u, elm ) | ( operation ) |
The first variant returns the normalizer of the finitely generated subgroup u2 in u1.
The second variant returns the normalizer of ⟨ elm ⟩ in the finitely generated subgroup u (see Normalizer (Reference: Normalizer) in the Reference Manual).
‣ RepresentativeAction( u, d, e ) | ( operation ) |
‣ IsConjugate( u, d, e ) | ( operation ) |
RepresentativeAction returns an element r ∈ u, where u is a finitely generated subgroup of a free group, such that d^r=e, or fail, if no such r exists. d and e may be elements or subgroups of u.
IsConjugate returns a boolean indicating whether such an element r exists.
‣ Centralizer( u, u2 ) | ( operation ) |
‣ Centralizer( u, elm ) | ( operation ) |
returns the centralizer of u2 or elm in the finitely generated subgroup u of a free group.
‣ Index( u1, u2 ) | ( operation ) |
‣ IndexNC( u1, u2 ) | ( operation ) |
return the index of u2 in u1, where u1 and u2 are finitely generated subgroups of a free group. The first variant returns fail if u2 is not a subgroup of u1, the second may return anything in this case.
‣ Intersection( u1, u2, \dots ) | ( function ) |
returns the intersection of u1 and u2, where u1 and u2 are finitely generated subgroups of a free group.
2.2-6 \in‣ \in( elm, u ) | ( method ) |
tests whether elm is contained in the finitely generated subgroup u of a free group.
‣ IsSubgroup( u1, u2 ) | ( function ) |
tests whether u2 is a subgroup of u1, where u1 and u2 are finitely generated subgroups of a free group.
2.2-8 \=‣ \=( u1, u2 ) | ( method ) |
test whether the two finitely generated subgroups u1 and u2 of a free group are equal.
‣ MinimalGeneratingSet( u ) | ( attribute ) |
‣ SmallGeneratingSet( u ) | ( attribute ) |
‣ GeneratorsOfGroup( u ) | ( attribute ) |
return generating sets for the finitely generated subgroup u of a free group. MinimalGeneratingSet and SmallGeneratingSet return the same free generators as FreeGeneratorsOfGroup, which are in fact a minimal generating set. GeneratorsOfGroup also returns these generators, if no other generators were stored at creation time.
It is not only possible to test whether an element is in a finitely generated subgroup of free group, this can also be done constructively. The idiomatic way to do so is by using a homomorphism.
Here is an example that computes how to write f.1^2 in the generators a=f1^2*f2^2 and b=f.1^2*f.2, checks the result, and then tries to write f.1 in the same generators:
gap> f := FreeGroup( 2 ); <free group on the generators [ f1, f2 ]> gap> ua := f.1^2*f.2^2;; ub := f.1^2*f.2;; gap> u := Group( ua, ub );; gap> g := FreeGroup( "a", "b" );; gap> hom := GroupHomomorphismByImages( g, u, > GeneratorsOfGroup(g), > GeneratorsOfGroup(u) ); [ a, b ] -> [ f1^2*f2^2, f1^2*f2 ] gap> # how can f.1^2 be expressed? gap> PreImagesRepresentative( hom, f.1^2 ); b*a^-1*b gap> last ^ hom; # check this f1^2 gap> ub * ua^-1 * ub; # another check f1^2 gap> PreImagesRepresentative( hom, f.1 ); # try f.1 fail gap> f.1 in u; false
There are also lower level operations to get the same results.
‣ AsWordLetterRepInGenerators( elm, u ) | ( operation ) |
‣ AsWordLetterRepInFreeGenerators( elm, u ) | ( operation ) |
return a letter representation (see Section Reference: Representations for Associative Words in the GAP Reference Manual) of the given elm relative to the generators the group was created with or the free generators as returned by FreeGeneratorsOfGroup.
Continuing the above example:
gap> AsWordLetterRepInGenerators( f.1^2, u ); [ 2, -1, 2 ] gap> AsWordLetterRepInFreeGenerators( f.1^2, u ); [ 2 ]
This means: to get f.1^2, multiply the second of the given generators with the inverse of the first and again with the second; or just take the second free generator.
The FGA package knows presentations of the automorphism groups of free groups. It also allows to express an automorphism as word in the generators of these presentations. This sections repeats the GAP standard methods to do so and shows functions to obtain the generating automorphisms.
‣ AutomorphismGroup( u ) | ( attribute ) |
returns the automorphism group of the finitely generated subgroup u of a free group.
Only a few methods will work with this group. But there is a way to obtain an isomorphic finitely presented group:
‣ IsomorphismFpGroup( group ) | ( attribute ) |
returns an isomorphism of group to a finitely presented group. For automorphism groups of free groups, the FGA package implements the presentations of [Neu33]. The finitely presented group itself can then be obtained with the command Range.
Here is an example:
gap> f := FreeGroup( 2 );; gap> a := AutomorphismGroup( f );; gap> iso := IsomorphismFpGroup( a );; gap> Range( iso ); <fp group on the generators [ O, P, U ]>
To express an automorphism as word in the generators of the presentation, just apply the isomorphism obtained from IsomorphismFpGroup.
gap> aut := GroupHomomorphismByImages( f, f, > GeneratorsOfGroup( f ), [ f.1^f.2, f.1*f.2 ] ); [ f1, f2 ] -> [ f2^-1*f1*f2, f1*f2 ] gap> ImageElm( iso, aut ); O^2*U*O*P^-1*U
It is also possible to use aut^iso or Image( iso, aut ). Using Image will perform additional checks on the arguments.
The FGA package provides a simpler way to create endomorphisms:
‣ FreeGroupEndomorphismByImages( g, imgs ) | ( function ) |
returns the endomorphism that maps the free generators of the finitely generated subgroup g of a free group to the elements listed in imgs. You may then apply IsBijective to check whether it is an automorphism.
The following functions return automorphisms that correspond to the generators in the presentation:
‣ FreeGroupAutomorphismsGeneratorO( group ) | ( function ) |
‣ FreeGroupAutomorphismsGeneratorP( group ) | ( function ) |
‣ FreeGroupAutomorphismsGeneratorU( group ) | ( function ) |
‣ FreeGroupAutomorphismsGeneratorS( group ) | ( function ) |
‣ FreeGroupAutomorphismsGeneratorT( group ) | ( function ) |
‣ FreeGroupAutomorphismsGeneratorQ( group ) | ( function ) |
‣ FreeGroupAutomorphismsGeneratorR( group ) | ( function ) |
return the automorphism which maps the free generators [x_1, x_2, dots, x_n] of group to
[x_1^-1, x_2, dots, x_n] (n≥1)
[x_2, x_1, x_3, dots, x_n] (n≥2)
[x_1x_2, x_2, x_3, dots, x_n] (n≥2)
[x_2^-1, x_3^-1, dots, x_n^-1, x_1^-1] (n≥1)
[x_2, x_1^-1, x_3, dots, x_n] (n≥2)
[x_2, x_3, dots, x_n, x_1] (n≥2)
[x_2^-1, x_1, x_3, x_4, dots, x_n-2, x_nx_n-1^-1, x_n-1^-1] (n≥4)
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For other non-standard options please see Chapter Reference: Installing a GAP Package in the GAP Reference Manual.
The FGA package is configured to autoload, so its functionality is usually available once GAP is started.
If GAP does not autoload, you can request the package with the LoadPackage command like this:
gap> LoadPackage( "fga" ); ----------------------------------------------------------------------------- Loading FGA 1.5.0-DEV (Free Group Algorithms) by Christian Sievers (c.sievers@tu-bs.de). maintained by: The GAP Team (support@gap-system.org). Homepage: https://gap-packages.github.io/fga/ Report issues at https://github.com/gap-packages/fga/issues ----------------------------------------------------------------------------- true
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fga-1.5.0/doc/chap1_mj.html 000644 000766 000024 00000022140 14413016355 015657 0 ustar 00mhorn staff 000000 000000This manual describes the FGA (Free Group Algorithms) package, a GAP package for computations with finitely generated subgroups of free groups.
This package allows you to (constructively) test membership and conjugacy, and to compute free generators, the rank, the index, normalizers, centralizers, and intersections where the groups involved are finitely generated subgroups of free groups. In addition, it provides generators and a finite presentation for the automorphism group of a finitely generated free group and allows to write any such automorphism as word in these generators.
See Chapter Functionality of the FGA package for details.
Chapter Installing and loading the FGA package explains how to install and load the FGA package.
The methods which are used work mainly with inverse finite automata, a variation of an idea known from theoretical computer science. An inverse finite automaton is a finite state automaton over a symmetric alphabet, i.e. one in which every letter has an inverse, such that each transition between two states for a letter corresponds to a transition in the opposite direction for the inverse letter.
Most of these techniques are described in Chapter 4 of [Sim94], where the same concept is called coset automaton. The method to obtain this automaton is called basic coset enumeration, and in fact it is coset enumeration where only important cosets are defined. Here a coset Gg is called important when there are words w and v such that wv is reduced and denotes an element of G and w denotes an element of Gg.
In [BMMW00], the connection between finitely generated subgroups of free groups and inverse finite automata is used to transfer results about the space complexity of problems concerning inverse finite automata to analogous results about finitely generated subgroups of free groups.
Chapter 6 of [Sim94] describes the Reidemeister-Schreier procedure and a variant called extended coset enumeration which yields a presentation in the given generators. The FGA package uses a variation thereof for its constructive membership test: it leaves out the part of the algorithm that fills in relations and interprets the resulting extended coset table differently. This algorithm might be called extended basic coset enumeration.
Some word oriented algorithms in the FGA package use basic facts about free groups. These can, for example, be found in [LS77].
The presentation of the automorphism groups follows [Neu33]. The algorithm for writing an automorphism in the generators works first at the level of Nielsen generators and uses relations from [Nie24].
The theoretical background for most of this implementation is explained in [Sie03].
The FGA package mainly installs new methods for operations that are already known to GAP. They overlap with methods in the GAP library in the case of groups of finite index. In this case, GAPs methods are usually faster, and the FGA package tries to recognize such cases and to refer to GAP.
The methods of the FGA package will only be selected when the groups involved know they are finitely generated. This may not always be the case for groups that were not created by methods of the FGA package. In such a case you will get a no method found error, or GAP may try a coset enumeration that stops with the message the coset enumeration has defined more than 256000 cosets. You may then call GeneratorsOfGroup, and try again.
Please inform the package author if you observe any remaining problems.
Like the GAP system itself, the FGA 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 package 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 can find the GNU General Public License in the file COPYING of the FGA package, and also in the file GPL in the etc directory of the main GAP distribution, or see http://www.gnu.org/licenses/gpl.html.
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fga-1.5.0/doc/chap3_mj.html 000644 000766 000024 00000011436 14413016355 015667 0 ustar 00mhorn staff 000000 000000The installation of the FGA package follows standard GAP rules. So the standard method is to unpack the archive into the pkg directory of your GAP distribution. This will create an fga subdirectory.
For other non-standard options please see Chapter Reference: Installing a GAP Package in the GAP Reference Manual.
The FGA package is configured to autoload, so its functionality is usually available once GAP is started.
If GAP does not autoload, you can request the package with the LoadPackage command like this:
gap> LoadPackage( "fga" ); ----------------------------------------------------------------------------- Loading FGA 1.5.0-DEV (Free Group Algorithms) by Christian Sievers (c.sievers@tu-bs.de). maintained by: The GAP Team (support@gap-system.org). Homepage: https://gap-packages.github.io/fga/ Report issues at https://github.com/gap-packages/fga/issues ----------------------------------------------------------------------------- true
You will not see the banner if FGA has already been loaded.
The LoadPackage command and ways to disable autoloading are described in Section Reference: Loading a GAP Package in the GAP Reference Manual.
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In the following, let f be a free group created by FreeGroup(n), and let u, u1 and u2 be finitely generated subgroups of f created by Group or Subgroup, or computed from some other subgroup of f. Let elm be an element of f.
For example:
gap> f := FreeGroup( 2 ); <free group on the generators [ f1, f2 ]> gap> u := Group( f.1^2, f.2^2, f.1*f.2 ); Group([ f1^2, f2^2, f1*f2 ]) gap> u1 := Subgroup( u, [f.1^2, f.1^4*f.2^6] ); Group([ f1^2, f1^4*f2^6 ]) gap> elm := f.1; f1 gap> u2 := Normalizer( u, elm ); Group([ f1^2 ])
These new operations are available for finitely generated subgroups of free groups:
‣ FreeGeneratorsOfGroup( u ) | ( attribute ) |
returns a list of free generators of the finitely generated subgroup u of a free group.
The elements in this list form an N-reduced set. In addition to being a free (and thus minimal) generating set for u, this means that whenever v1, v2 and v3 are elements or inverses of elements of this list, then
\(\textit{v1}\textit{v2}\neq1\) implies \(|\textit{v1}\textit{v2}|\geq\max(|\textit{v1}|, |\textit{v2}|)\), and
\(\textit{v1}\textit{v2}\neq1\) and \(\textit{v2}\textit{v3}\neq1\) implies \(|\textit{v1}\textit{v2}\textit{v3}| > |\textit{v1}| - |\textit{v2}| + |\textit{v3}|\)
hold, where \(|.|\) denotes the word length.
‣ RankOfFreeGroup( u ) | ( attribute ) |
‣ Rank( u ) | ( operation ) |
returns the rank of the finitely generated subgroup u of a free group.
‣ CyclicallyReducedWord( elm ) | ( operation ) |
returns the cyclically reduced form of elm.
This section lists operations that are already known to GAP. FGA installs new methods for them so that they can also be used with free groups. In cases where FGA installs methods that are usually only used internally, user functions are shown instead.
‣ Normalizer( u1, u2 ) | ( operation ) |
‣ Normalizer( u, elm ) | ( operation ) |
The first variant returns the normalizer of the finitely generated subgroup u2 in u1.
The second variant returns the normalizer of \(\langle \textit{elm} \rangle\) in the finitely generated subgroup u (see Normalizer (Reference: Normalizer) in the Reference Manual).
‣ RepresentativeAction( u, d, e ) | ( operation ) |
‣ IsConjugate( u, d, e ) | ( operation ) |
RepresentativeAction returns an element \( \textit{r} \in \textit{u} \), where u is a finitely generated subgroup of a free group, such that \(\textit{d}^{\textit{r}}=\textit{e}\), or fail, if no such r exists. d and e may be elements or subgroups of u.
IsConjugate returns a boolean indicating whether such an element r exists.
‣ Centralizer( u, u2 ) | ( operation ) |
‣ Centralizer( u, elm ) | ( operation ) |
returns the centralizer of u2 or elm in the finitely generated subgroup u of a free group.
‣ Index( u1, u2 ) | ( operation ) |
‣ IndexNC( u1, u2 ) | ( operation ) |
return the index of u2 in u1, where u1 and u2 are finitely generated subgroups of a free group. The first variant returns fail if u2 is not a subgroup of u1, the second may return anything in this case.
‣ Intersection( u1, u2, \dots ) | ( function ) |
returns the intersection of u1 and u2, where u1 and u2 are finitely generated subgroups of a free group.
2.2-6 \in‣ \in( elm, u ) | ( method ) |
tests whether elm is contained in the finitely generated subgroup u of a free group.
‣ IsSubgroup( u1, u2 ) | ( function ) |
tests whether u2 is a subgroup of u1, where u1 and u2 are finitely generated subgroups of a free group.
2.2-8 \=‣ \=( u1, u2 ) | ( method ) |
test whether the two finitely generated subgroups u1 and u2 of a free group are equal.
‣ MinimalGeneratingSet( u ) | ( attribute ) |
‣ SmallGeneratingSet( u ) | ( attribute ) |
‣ GeneratorsOfGroup( u ) | ( attribute ) |
return generating sets for the finitely generated subgroup u of a free group. MinimalGeneratingSet and SmallGeneratingSet return the same free generators as FreeGeneratorsOfGroup, which are in fact a minimal generating set. GeneratorsOfGroup also returns these generators, if no other generators were stored at creation time.
It is not only possible to test whether an element is in a finitely generated subgroup of free group, this can also be done constructively. The idiomatic way to do so is by using a homomorphism.
Here is an example that computes how to write f.1^2 in the generators a=f1^2*f2^2 and b=f.1^2*f.2, checks the result, and then tries to write f.1 in the same generators:
gap> f := FreeGroup( 2 ); <free group on the generators [ f1, f2 ]> gap> ua := f.1^2*f.2^2;; ub := f.1^2*f.2;; gap> u := Group( ua, ub );; gap> g := FreeGroup( "a", "b" );; gap> hom := GroupHomomorphismByImages( g, u, > GeneratorsOfGroup(g), > GeneratorsOfGroup(u) ); [ a, b ] -> [ f1^2*f2^2, f1^2*f2 ] gap> # how can f.1^2 be expressed? gap> PreImagesRepresentative( hom, f.1^2 ); b*a^-1*b gap> last ^ hom; # check this f1^2 gap> ub * ua^-1 * ub; # another check f1^2 gap> PreImagesRepresentative( hom, f.1 ); # try f.1 fail gap> f.1 in u; false
There are also lower level operations to get the same results.
‣ AsWordLetterRepInGenerators( elm, u ) | ( operation ) |
‣ AsWordLetterRepInFreeGenerators( elm, u ) | ( operation ) |
return a letter representation (see Section Reference: Representations for Associative Words in the GAP Reference Manual) of the given elm relative to the generators the group was created with or the free generators as returned by FreeGeneratorsOfGroup.
Continuing the above example:
gap> AsWordLetterRepInGenerators( f.1^2, u ); [ 2, -1, 2 ] gap> AsWordLetterRepInFreeGenerators( f.1^2, u ); [ 2 ]
This means: to get f.1^2, multiply the second of the given generators with the inverse of the first and again with the second; or just take the second free generator.
The FGA package knows presentations of the automorphism groups of free groups. It also allows to express an automorphism as word in the generators of these presentations. This sections repeats the GAP standard methods to do so and shows functions to obtain the generating automorphisms.
‣ AutomorphismGroup( u ) | ( attribute ) |
returns the automorphism group of the finitely generated subgroup u of a free group.
Only a few methods will work with this group. But there is a way to obtain an isomorphic finitely presented group:
‣ IsomorphismFpGroup( group ) | ( attribute ) |
returns an isomorphism of group to a finitely presented group. For automorphism groups of free groups, the FGA package implements the presentations of [Neu33]. The finitely presented group itself can then be obtained with the command Range.
Here is an example:
gap> f := FreeGroup( 2 );; gap> a := AutomorphismGroup( f );; gap> iso := IsomorphismFpGroup( a );; gap> Range( iso ); <fp group on the generators [ O, P, U ]>
To express an automorphism as word in the generators of the presentation, just apply the isomorphism obtained from IsomorphismFpGroup.
gap> aut := GroupHomomorphismByImages( f, f, > GeneratorsOfGroup( f ), [ f.1^f.2, f.1*f.2 ] ); [ f1, f2 ] -> [ f2^-1*f1*f2, f1*f2 ] gap> ImageElm( iso, aut ); O^2*U*O*P^-1*U
It is also possible to use aut^iso or Image( iso, aut ). Using Image will perform additional checks on the arguments.
The FGA package provides a simpler way to create endomorphisms:
‣ FreeGroupEndomorphismByImages( g, imgs ) | ( function ) |
returns the endomorphism that maps the free generators of the finitely generated subgroup g of a free group to the elements listed in imgs. You may then apply IsBijective to check whether it is an automorphism.
The following functions return automorphisms that correspond to the generators in the presentation:
‣ FreeGroupAutomorphismsGeneratorO( group ) | ( function ) |
‣ FreeGroupAutomorphismsGeneratorP( group ) | ( function ) |
‣ FreeGroupAutomorphismsGeneratorU( group ) | ( function ) |
‣ FreeGroupAutomorphismsGeneratorS( group ) | ( function ) |
‣ FreeGroupAutomorphismsGeneratorT( group ) | ( function ) |
‣ FreeGroupAutomorphismsGeneratorQ( group ) | ( function ) |
‣ FreeGroupAutomorphismsGeneratorR( group ) | ( function ) |
return the automorphism which maps the free generators [\( x_1, x_2, \dots, x_n \)] of group to
[\( x_1^{-1}, x_2, \dots, x_n \)] (\(n\ge1\))
[\( x_2, x_1, x_3, \dots, x_n \)] (\(n\ge2\))
[\( x_1x_2, x_2, x_3, \dots, x_n \)] (\(n\ge2\))
[\( x_2^{-1}, x_3^{-1}, \dots, x_n^{-1}, x_1^{-1} \)] (\(n\ge1\))
[\( x_2, x_1^{-1}, x_3, \dots, x_n \)] (\(n\ge2\))
[\( x_2, x_3, \dots, x_n, x_1 \)] (\(n\ge2\))
[\( x_2^{-1}, x_1, x_3, x_4, \dots, x_{n-2}, x_nx_{n-1}^{-1}, x_{n-1}^{-1} \)] (\(n\ge4\))
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Nielsen}, title = {Die {I}somorphismengruppe der freien {G}ruppen}, journal = {Math. Annalen}, volume = 91, year = 1924, pages = {169--209} } @book{Sims94, Author = "C.~C. 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[[20XLS77[120X] [16XLyndon, R. C. and Schupp, P. E.[116X, [17XCombinatorial Group Theory[117X, Springer (1977). [[20XNeu33[120X] [16XNeumann, B.[116X, [17XDie Automorphismengruppe der freien Gruppen[117X, [18XMath. Annalen[118X, [19X107[119X (1933), 367--386. [[20XNie24[120X] [16XNielsen, J.[116X, [17XDie Isomorphismengruppe der freien Gruppen[117X, [18XMath. Annalen[118X, [19X91[119X (1924), 169--209. [[20XSie03[120X] [16XSievers, C.[116X, [17XAlgorithmen f\accent127ur freie Gruppen[117X, Diplomarbeit, TU Braunschweig (2003). [[20XSim94[120X] [16XSims, C. C.[116X, [17XComputation with Finitely Presented Groups[117X, Cambridge University Press, Encyclopedia of Mathematics and its Applications, [19X48[119X, Cambridge (1994), xiii+604 pages. [32X fga-1.5.0/doc/rainbow.js 000644 000766 000024 00000005336 14413016355 015316 0 ustar 00mhorn staff 000000 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'); fga-1.5.0/doc/FGA.xml 000644 000766 000024 00000000623 14413016350 014423 0 ustar 00mhorn staff 000000 000000 ] >[BMMW00] Birget, J. -.C., Margolis, S., Meakin, J. and Weil, P., PSPACE-complete problems for subgroups of free groups and inverse finite automata, Theoretical Computer Science, 242 (2000), 247--281.
[LS77] Lyndon, R. C. and Schupp, P. E., Combinatorial Group Theory, Springer (1977).
[Neu33] Neumann, B., Die Automorphismengruppe der freien Gruppen, Math. Annalen, 107 (1933), 367--386.
[Nie24] Nielsen, J., Die Isomorphismengruppe der freien Gruppen, Math. Annalen, 91 (1924), 169--209.
[Sie03] Sievers, C., Algorithmen f\accent127ur freie Gruppen, Diplomarbeit, TU Braunschweig (2003).
[Sim94] Sims, C. C., Computation with Finitely Presented Groups, Cambridge University Press, Encyclopedia of Mathematics and its Applications, 48, Cambridge (1994), xiii+604 pages.
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fga-1.5.0/doc/chap0.html 000644 000766 000024 00000016246 14413016355 015202 0 ustar 00mhorn staff 000000 0000001.5.0
4 April 2023
Note: This version of FGA is a fork of the original FGA, maintained by the GAP Team.
Christian Sievers
Email: c.sievers@tu-bs.de
generated by GAPDoc2HTML
fga-1.5.0/doc/chap1.txt 000644 000766 000024 00000012611 14413016350 015041 0 ustar 00mhorn staff 000000 000000 [1X1 [33X[0;0YIntroduction[133X[101X [1X1.1 [33X[0;0YOverview[133X[101X [33X[0;0YThis manual describes the [5XFGA[105X ([13XFree Group Algorithms[113X) package, a [5XGAP[105X package for computations with finitely generated subgroups of free groups.[133X [33X[0;0YThis package allows you to (constructively) test membership and conjugacy, and to compute free generators, the rank, the index, normalizers, centralizers, and intersections where the groups involved are finitely generated subgroups of free groups. In addition, it provides generators and a finite presentation for the automorphism group of a finitely generated free group and allows to write any such automorphism as word in these generators.[133X [33X[0;0YSee Chapter [14X'[33X[0;0YFunctionality of the FGA package[133X'[114X for details.[133X [33X[0;0YChapter [14X'[33X[0;0YInstalling and loading the FGA package[133X'[114X explains how to install and load the [5XFGA[105X package.[133X [1X1.2 [33X[0;0YImplementation and background[133X[101X [33X[0;0YThe methods which are used work mainly with inverse finite automata, a variation of an idea known from theoretical computer science. An inverse finite automaton is a finite state automaton over a symmetric alphabet, i.e. one in which every letter has an inverse, such that each transition between two states for a letter corresponds to a transition in the opposite direction for the inverse letter.[133X [33X[0;0YMost of these techniques are described in Chapter 4 of [Sim94], where the same concept is called coset automaton. The method to obtain this automaton is called basic coset enumeration, and in fact it is coset enumeration where only important cosets are defined. Here a coset [3XGg[103X is called important when there are words [3Xw[103X and [3Xv[103X such that [3Xwv[103X is reduced and denotes an element of [3XG[103X and [3Xw[103X denotes an element of [3XGg[103X.[133X [33X[0;0YIn [BMMW00], the connection between finitely generated subgroups of free groups and inverse finite automata is used to transfer results about the space complexity of problems concerning inverse finite automata to analogous results about finitely generated subgroups of free groups.[133X [33X[0;0YChapter 6 of [Sim94] describes the Reidemeister-Schreier procedure and a variant called extended coset enumeration which yields a presentation in the given generators. The [5XFGA[105X package uses a variation thereof for its constructive membership test: it leaves out the part of the algorithm that fills in relations and interprets the resulting extended coset table differently. This algorithm might be called extended basic coset enumeration.[133X [33X[0;0YSome word oriented algorithms in the [5XFGA[105X package use basic facts about free groups. These can, for example, be found in [LS77].[133X [33X[0;0YThe presentation of the automorphism groups follows [Neu33]. The algorithm for writing an automorphism in the generators works first at the level of Nielsen generators and uses relations from [Nie24].[133X [33X[0;0YThe theoretical background for most of this implementation is explained in [Sie03].[133X [1X1.3 [33X[0;0YIntegration of the package[133X[101X [33X[0;0YThe [5XFGA[105X package mainly installs new methods for operations that are already known to [5XGAP[105X. They overlap with methods in the [5XGAP[105X library in the case of groups of finite index. In this case, [5XGAP[105Xs methods are usually faster, and the [5XFGA[105X package tries to recognize such cases and to refer to [5XGAP[105X.[133X [33X[0;0YThe methods of the [5XFGA[105X package will only be selected when the groups involved know they are finitely generated. This may not always be the case for groups that were not created by methods of the [5XFGA[105X package. In such a case you will get a [10Xno method found[110X error, or [5XGAP[105X may try a coset enumeration that stops with the message [10Xthe coset enumeration has defined more than 256000 cosets[110X. You may then call [10XGeneratorsOfGroup[110X, and try again.[133X [33X[0;0YPlease inform the package author if you observe any remaining problems.[133X [1X1.4 [33X[0;0YLicense[133X[101X [33X[0;0YLike the [5XGAP[105X system itself, the [5XFGA[105X 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.[133X [33X[0;0YThis package 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.[133X [33X[0;0YYou can find the GNU General Public License in the file [10XCOPYING[110X of the [5XFGA[105X package, and also in the file [10XGPL[110X in the [10Xetc[110X directory of the main [5XGAP[105X distribution, or see [7Xhttp://www.gnu.org/licenses/gpl.html[107X.[133X fga-1.5.0/doc/chap0.txt 000644 000766 000024 00000003334 14413016350 015042 0 ustar 00mhorn staff 000000 000000 [1X FGA [101X [1X Free Group Algorithms [101X 1.5.0 4 April 2023 Christian Sievers [33X[0;10Y[13XNote:[113X This version of FGA is a fork of the original FGA, maintained by the GAP Team.[133X Christian Sievers Email: [7Xmailto:c.sievers@tu-bs.de[107X ------------------------------------------------------- [1XContents (FGA)[101X 1 [33X[0;0YIntroduction[133X 1.1 [33X[0;0YOverview[133X 1.2 [33X[0;0YImplementation and background[133X 1.3 [33X[0;0YIntegration of the package[133X 1.4 [33X[0;0YLicense[133X 2 [33X[0;0YFunctionality of the FGA package[133X 2.1 [33X[0;0YNew operations for free groups[133X 2.1-1 FreeGeneratorsOfGroup 2.1-2 RankOfFreeGroup 2.1-3 CyclicallyReducedWord 2.2 [33X[0;0YMethod installations[133X 2.2-1 Normalizer 2.2-2 RepresentativeAction 2.2-3 Centralizer 2.2-4 Index 2.2-5 Intersection 2.2-6 \in 2.2-7 IsSubgroup 2.2-8 \= 2.2-9 MinimalGeneratingSet 2.3 [33X[0;0YConstructive membership test[133X 2.3-1 AsWordLetterRepInGenerators 2.4 [33X[0;0YAutomorphism groups of free groups[133X 2.4-1 AutomorphismGroup 2.4-2 IsomorphismFpGroup 2.4-3 FreeGroupEndomorphismByImages 2.4-4 FreeGroupAutomorphismsGeneratorO 3 [33X[0;0YInstalling and loading the FGA package[133X 3.1 [33X[0;0YInstalling the FGA package[133X 3.2 [33X[0;0YLoading the FGA package[133X [32X fga-1.5.0/doc/manual.css 000644 000766 000024 00000015754 14413016355 015313 0 ustar 00mhorn staff 000000 000000 /* manual.css Frank Lübeck */ /* This is the default CSS style sheet for GAPDoc HTML manuals. */ /* basic settings, fonts, sizes, colors, ... */ body { position: relative; 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} fga-1.5.0/doc/chap1.html 000644 000766 000024 00000021555 14413016355 015202 0 ustar 00mhorn staff 000000 000000This manual describes the FGA (Free Group Algorithms) package, a GAP package for computations with finitely generated subgroups of free groups.
This package allows you to (constructively) test membership and conjugacy, and to compute free generators, the rank, the index, normalizers, centralizers, and intersections where the groups involved are finitely generated subgroups of free groups. In addition, it provides generators and a finite presentation for the automorphism group of a finitely generated free group and allows to write any such automorphism as word in these generators.
See Chapter Functionality of the FGA package for details.
Chapter Installing and loading the FGA package explains how to install and load the FGA package.
The methods which are used work mainly with inverse finite automata, a variation of an idea known from theoretical computer science. An inverse finite automaton is a finite state automaton over a symmetric alphabet, i.e. one in which every letter has an inverse, such that each transition between two states for a letter corresponds to a transition in the opposite direction for the inverse letter.
Most of these techniques are described in Chapter 4 of [Sim94], where the same concept is called coset automaton. The method to obtain this automaton is called basic coset enumeration, and in fact it is coset enumeration where only important cosets are defined. Here a coset Gg is called important when there are words w and v such that wv is reduced and denotes an element of G and w denotes an element of Gg.
In [BMMW00], the connection between finitely generated subgroups of free groups and inverse finite automata is used to transfer results about the space complexity of problems concerning inverse finite automata to analogous results about finitely generated subgroups of free groups.
Chapter 6 of [Sim94] describes the Reidemeister-Schreier procedure and a variant called extended coset enumeration which yields a presentation in the given generators. The FGA package uses a variation thereof for its constructive membership test: it leaves out the part of the algorithm that fills in relations and interprets the resulting extended coset table differently. This algorithm might be called extended basic coset enumeration.
Some word oriented algorithms in the FGA package use basic facts about free groups. These can, for example, be found in [LS77].
The presentation of the automorphism groups follows [Neu33]. The algorithm for writing an automorphism in the generators works first at the level of Nielsen generators and uses relations from [Nie24].
The theoretical background for most of this implementation is explained in [Sie03].
The FGA package mainly installs new methods for operations that are already known to GAP. They overlap with methods in the GAP library in the case of groups of finite index. In this case, GAPs methods are usually faster, and the FGA package tries to recognize such cases and to refer to GAP.
The methods of the FGA package will only be selected when the groups involved know they are finitely generated. This may not always be the case for groups that were not created by methods of the FGA package. In such a case you will get a no method found error, or GAP may try a coset enumeration that stops with the message the coset enumeration has defined more than 256000 cosets. You may then call GeneratorsOfGroup, and try again.
Please inform the package author if you observe any remaining problems.
Like the GAP system itself, the FGA 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 package 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 can find the GNU General Public License in the file COPYING of the FGA package, and also in the file GPL in the etc directory of the main GAP distribution, or see http://www.gnu.org/licenses/gpl.html.
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