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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Alexander Hulpke.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
############################################################################
##
#R IsFromFpGroupGeneralMapping(<map>)
#R IsFromFpGroupHomomorphism(<map>)
##
## <ManSection>
## <Filt Name="IsFromFpGroupGeneralMapping" Arg='map' Type='Representation'/>
## <Filt Name="IsFromFpGroupHomomorphism" Arg='map' Type='Representation'/>
##
## <Description>
## is the representation of mappings from an fp group.
## </Description>
## </ManSection>
##
DeclareCategory( "IsFromFpGroupGeneralMapping", IsGroupGeneralMapping
# we want all methods for homs from fp groups to be better. This (slight
# hack) increases the rank of the category of such mappings.
and NewFilter("Extrarankfilter",10));
DeclareSynonym("IsFromFpGroupHomomorphism",
IsFromFpGroupGeneralMapping and IsMapping);
############################################################################
##
#R IsFromFpGroupGeneralMappingByImages(<map>)
#R IsFromFpGroupHomomorphismByImages(<map>)
##
## <#GAPDoc Label="IsFromFpGroupGeneralMappingByImages">
## <ManSection>
## <Filt Name="IsFromFpGroupGeneralMappingByImages" Arg='map'
## Type='Representation'/>
## <Filt Name="IsFromFpGroupHomomorphismByImages" Arg='map'
## Type='Representation'/>
##
## <Description>
## is the representation of mappings from an fp group.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareRepresentation( "IsFromFpGroupGeneralMappingByImages",
IsFromFpGroupGeneralMapping and IsGroupGeneralMappingByImages,
[ "generators", "genimages" ] );
DeclareSynonym("IsFromFpGroupHomomorphismByImages",
IsFromFpGroupGeneralMappingByImages and IsMapping);
############################################################################
##
#R IsFromFpGroupStdGensGeneralMappingByImages(<map>)
#R IsFromFpGroupStdGensHomomorphismByImages(<map>)
##
## <#GAPDoc Label="IsFromFpGroupStdGensGeneralMappingByImages">
## <ManSection>
## <Filt Name="IsFromFpGroupStdGensGeneralMappingByImages" Arg='map'
## Type='Representation'/>
## <Filt Name="IsFromFpGroupStdGensHomomorphismByImages" Arg='map'
## Type='Representation'/>
##
## <Description>
## is the representation of total mappings from an fp group that give images of
## the standard generators.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareRepresentation( "IsFromFpGroupStdGensGeneralMappingByImages",
IsFromFpGroupGeneralMappingByImages, [ "generators", "genimages" ] );
DeclareSynonym("IsFromFpGroupStdGensHomomorphismByImages",
IsFromFpGroupStdGensGeneralMappingByImages and IsMapping);
############################################################################
##
#R IsToFpGroupGeneralMappingByImages(<map>)
#R IsToFpGroupHomomorphismByImages(<map>)
##
## <ManSection>
## <Filt Name="IsToFpGroupGeneralMappingByImages" Arg='map' Type='Representation'/>
## <Filt Name="IsToFpGroupHomomorphismByImages" Arg='map' Type='Representation'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareRepresentation( "IsToFpGroupGeneralMappingByImages",
IsGroupGeneralMappingByImages,
[ "generators", "genimages" ] );
DeclareSynonym("IsToFpGroupHomomorphismByImages",
IsToFpGroupGeneralMappingByImages and IsMapping);
############################################################################
##
#P IsWordDecompHomomorphism(<map>)
##
## <ManSection>
## <Prop Name="IsWordDecompHomomorphism" Arg='map'/>
##
## <Description>
## these homomorphism contain a component <C>!.decompinfo</C> that provides
## functionality to decompose a word into generators. They are primarily
## used for <C>IsomorphismFpGroupBy...Series</C>.
## </Description>
## </ManSection>
##
DeclareProperty( "IsWordDecompHomomorphism",IsGroupGeneralMappingByImages);
#############################################################################
##
#A CosetTableFpHom(<hom>)
##
## <ManSection>
## <Attr Name="CosetTableFpHom" Arg='hom'/>
##
## <Description>
## returns an augmented coset table for an homomorphism from an fp group,
## corresponding to the !.generators component. The component
## <C>.secondaryImages</C> of this table will give the images of all (primary
## and secondary) subgroup generators under <A>hom</A>.
## <P/>
## As we might want to add further entries to the table, it is a mutable
## attribute.
## </Description>
## </ManSection>
##
DeclareAttribute("CosetTableFpHom",IsGeneralMapping,"mutable");
#############################################################################
##
#F SecondaryImagesAugmentedCosetTable(<aug>,<gens>,<genimages>)
##
## <ManSection>
## <Func Name="SecondaryImagesAugmentedCosetTable" Arg='aug,gens,genimages'/>
##
## <Description>
## returns a list of images of the secondary generators, based on the
## components <C>homgens</C> and <C>homgenims</C> in the augmented coset table <A>aug</A>.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("SecondaryImagesAugmentedCosetTable");
#############################################################################
##
#F TrySecondaryImages(<aug>)
##
## <ManSection>
## <Func Name="TrySecondaryImages" Arg='aug'/>
##
## <Description>
## sets a component <C>secondaryImages</C> in the augmented coset table (seeded
## to a ShallowCopy of the primary images) if having all these images
## cannot become too memory extensive. (Call this function for augmented
## coset tables for homomorphisms once -- the other functions make use of
## the <C>secondaryImages</C> component if existing.)
## </Description>
## </ManSection>
##
DeclareGlobalFunction("TrySecondaryImages");
#############################################################################
##
#F KuKGenerators( <G>, <beta>, <alpha> )
##
## <#GAPDoc Label="KuKGenerators">
## <ManSection>
## <Func Name="KuKGenerators" Arg='G, beta, alpha'/>
##
## <Description>
## <Index>Krasner-Kaloujnine theorem</Index>
## <Index>Wreath product embedding</Index>
## If <A>beta</A> is a homomorphism from <A>G</A> into a transitive
## permutation group, <M>U</M> the full preimage of the point stabilizer and
## <A>alpha</A> a homomorphism defined on (a superset) of <M>U</M>,
## this function returns images of the generators of <A>G</A> when mapping
## to the wreath product <M>(U <A>alpha</A>) \wr (<A>G</A> <A>beta</A>)</M>.
## (This is the Krasner-Kaloujnine embedding theorem.)
## <Example><![CDATA[
## gap> g:=Group((1,2,3,4),(1,2));;
## gap> hom:=GroupHomomorphismByImages(g,Group((1,2)),
## > GeneratorsOfGroup(g),[(1,2),(1,2)]);;
## gap> u:=PreImage(hom,Stabilizer(Image(hom),1));
## Group([ (2,3,4), (1,2,4) ])
## gap> hom2:=GroupHomomorphismByImages(u,Group((1,2,3)),
## > GeneratorsOfGroup(u),[ (1,2,3), (1,2,3) ]);;
## gap> KuKGenerators(g,hom,hom2);
## [ (1,4)(2,5)(3,6), (1,6)(2,4)(3,5) ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("KuKGenerators");
#############################################################################
##
#A IsomorphismSimplifiedFpGroup( <G> )
##
## <#GAPDoc Label="IsomorphismSimplifiedFpGroup">
## <ManSection>
## <Attr Name="IsomorphismSimplifiedFpGroup" Arg='G'/>
##
## <Description>
## applies Tietze transformations to a copy of the presentation of the
## given finitely presented group <A>G</A> in order to reduce it
## with respect to the number of generators, the number of relators,
## and the relator lengths.
## <P/>
## The operation returns an isomorphism with source <A>G</A>, range a group
## <A>H</A> isomorphic to <A>G</A>, so that the presentation of <A>H</A> has
## been simplified using Tietze transformations.
## <Example><![CDATA[
## gap> f:=FreeGroup(3);;
## gap> g:=f/[f.1^2,f.2^3,(f.1*f.2)^5,f.1/f.3];
## <fp group on the generators [ f1, f2, f3 ]>
## gap> hom:=IsomorphismSimplifiedFpGroup(g);
## [ f1, f2, f3 ] -> [ f1, f2, f1 ]
## gap> Range(hom);
## <fp group on the generators [ f1, f2 ]>
## gap> RelatorsOfFpGroup(Range(hom));
## [ f1^2, f2^3, (f1*f2)^5 ]
## gap> RelatorsOfFpGroup(g);
## [ f1^2, f2^3, (f1*f2)^5, f1*f3^-1 ]
## ]]></Example>
## <P/>
## <Ref Attr="IsomorphismSimplifiedFpGroup"/> uses Tietze transformations
## to simplify the presentation, see <Ref Sect="SimplifiedFpGroup"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("IsomorphismSimplifiedFpGroup",IsSubgroupFpGroup);
#############################################################################
##
#A EpimorphismFromFreeGroup( <G> )
##
## <#GAPDoc Label="EpimorphismFromFreeGroup">
## <ManSection>
## <Attr Name="EpimorphismFromFreeGroup" Arg='G'/>
##
## <Description>
## For a group <A>G</A> with a known generating set, this attribute returns
## a homomorphism from a free group that maps the free generators to the
## generators of <A>G</A>.
## <P/>
## The option <C>names</C> can be used to prescribe a (print) name
## for the free generators.
## <P/>
## The following example shows how to decompose elements of <M>S_4</M> in the
## generators <C>(1,2,3,4)</C> and <C>(1,2)</C>:
## <P/>
## <Example><![CDATA[
## gap> g:=Group((1,2,3,4),(1,2));
## Group([ (1,2,3,4), (1,2) ])
## gap> hom:=EpimorphismFromFreeGroup(g:names:=["x","y"]);
## [ x, y ] -> [ (1,2,3,4), (1,2) ]
## gap> PreImagesRepresentative(hom,(1,4));
## y^-1*x^-1*(x^-1*y^-1)^2*x
## ]]></Example>
## <P/>
## The following example stems from a real request to the &GAP; Forum.
## In September 2000 a &GAP; user working with puzzles wanted to express the
## permutation <C>(1,2)</C> as a word as short as possible in particular
## generators of the symmetric group <M>S_{16}</M>.
## <P/>
## <Example><![CDATA[
## gap> perms := [ (1,2,3,7,11,10,9,5), (2,3,4,8,12,11,10,6),
## > (5,6,7,11,15,14,13,9), (6,7,8,12,16,15,14,10) ];;
## gap> puzzle := Group( perms );;Size( puzzle );
## 20922789888000
## gap> hom:=EpimorphismFromFreeGroup(puzzle:names:=["a", "b", "c", "d"]);;
## gap> word := PreImagesRepresentative( hom, (1,2) );
## a^-1*c*b*c^-1*a*b^-1*a^-2*c^-1*a*b^-1*c*b
## gap> Length( word );
## 13
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("EpimorphismFromFreeGroup",IsGroup);
#############################################################################
##
#F LargerQuotientBySubgroupAbelianization( <hom>,<U> )
##
## <#GAPDoc Label="LargerQuotientBySubgroupAbelianization">
## <ManSection>
## <Func Name="LargerQuotientBySubgroupAbelianization" Arg='hom, U'/>
##
## <Description>
## Let <A>hom</A> a homomorphism from a finitely presented group <M>G</M>
## to a finite group <M>H</M> and <M><A>U</A> \leq H</M>.
## This function will –if it exists– return a subgroup
## <M>S \leq <A>G</A></M>, such that the core of <M>S</M> is properly
## contained in the kernel of <A>hom</A> as well as in the derived subgroup
## of <M>V</M>,
## where <M>V</M> is the pre-image of <A>U</A> under <A>hom</A>.
## Thus <M>S</M> exposes a larger quotient of <M>G</M>.
## If no such subgroup exists, <K>fail</K> is returned.
## <Example><![CDATA[
## gap> f:=FreeGroup("x","y","z");;
## gap> g:=f/ParseRelators(f,"x^3=y^3=z^5=(xyx^2y^2)^2=(xz)^2=(yz^3)^2=1");
## <fp group on the generators [ x, y, z ]>
## gap> l:=LowIndexSubgroupsFpGroup(g,6);;
## gap> List(l,IndexInWholeGroup);
## [ 1, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6 ]
## gap> q:=DefiningQuotientHomomorphism(l[6]);;p:=Image(q);Size(p);
## Group([ (4,5,6), (1,2,3)(4,6,5), (2,4,6,3,5) ])
## 360
## gap> s:=LargerQuotientBySubgroupAbelianization(q,SylowSubgroup(p,3));
## Group(<fp, no generators known>)
## gap> Size(Image(DefiningQuotientHomomorphism(s)));
## 193273528320
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("LargerQuotientBySubgroupAbelianization");
#############################################################################
##
#F ProcessEpimorphismToNewFpGroup( <hom> )
##
## <#GAPDoc Label="ProcessHomomorphismToNewFpGroup">
## <ManSection>
## <Func Name="LargerQuotientBySubgroupAbelianization" Arg='hom'/>
##
## <Description>
## Let <A>hom</A> a homomorphism from a group <A>G</A> to a newly created
## finitely presented group <A>F</A> (in which no calculation yet has taken
## place) such that the inverse of <A>hom</A> can be applied to elements of
## $F$. This function endows <A>F</A> (or, more
## correctly, its elements family) with information (in the form of an
## <C>FPFaithHom</C>) about <A>G</A> that can be
## used to obtain a faithful representation for <A>F</A>. This can be in the
## form of <A>G</A> itself being a permutation/matrix or fp group. It also can
## be if <A>G</A> is finitely presented but (its family) has an
## <C>FPFaithHom</A>. (However it will not apply if <A>G</A> is a subgroup
## of an fp group.) The result of this is make higher level calculations in
## <A>F</A> more efficient.
## This is an internal function that does not test its arguments and the
## result is undefined if calling the function on any other object.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("ProcessEpimorphismToNewFpGroup");
#############################################################################
##
#M InducedRepFpGroup(<hom>,<u> )
##
## induce <hom> def. on <u> up to the full group
DeclareGlobalFunction("InducedRepFpGroup");
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