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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer, Alexander Hulpke, Heiko Theißen.
##
## 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
##
## 1. Functions for creating group general mappings by images
## 2. Functions for creating natural homomorphisms
## 3. Functions for conjugation action
## 4. Functions for ...
##
#############################################################################
##
## 1. Functions for creating group general mappings by images
##
#############################################################################
##
#O GroupGeneralMappingByImages( <G>, <H>, <gens>, <imgs> )
##
## <#GAPDoc Label="GroupGeneralMappingByImages">
## <ManSection>
## <Oper Name="GroupGeneralMappingByImages" Arg='G, H, gens, imgs'/>
## <Oper Name="GroupGeneralMappingByImages" Arg='G, gens, imgs' Label="from group to itself"/>
## <Oper Name="GroupGeneralMappingByImagesNC" Arg='G, H, gens, imgs'/>
## <Oper Name="GroupGeneralMappingByImagesNC" Arg='G, gens, imgs' Label="from group to itself"/>
##
## <Description>
## returns a general mapping defined by extending the mapping from
## <A>gens</A> to <A>imgs</A> homomorphically. If the range <A>H</A> is not
## given the mapping will be made automatically surjective. The NC version
## does not test whether <A>gens</A> are contained in <A>G</A> or <A>imgs</A>
## are contained in <A>H</A>.
## (<Ref Func="GroupHomomorphismByImages"/> creates
## a group general mapping by images and
## tests whether it is in <Ref Filt="IsMapping"/>.)
## <Example><![CDATA[
## gap> map:=GroupGeneralMappingByImages(g,h,gens,[(1,2,3),(1,2)]);
## [ (1,2,3,4), (1,2) ] -> [ (1,2,3), (1,2) ]
## gap> IsMapping(map);
## false
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "GroupGeneralMappingByImages",
[ IsGroup, IsGroup, IsList, IsList ] );
DeclareOperation( "GroupGeneralMappingByImages",
[ IsGroup, IsList, IsList ] );
DeclareOperation( "GroupGeneralMappingByImagesNC",
[ IsGroup, IsGroup, IsList, IsList ] );
DeclareOperation( "GroupGeneralMappingByImagesNC",
[ IsGroup, IsList, IsList ] );
#############################################################################
##
#F GroupHomomorphismByImages( <G>[, <H>][[, <gens>], <imgs>] )
##
## <#GAPDoc Label="GroupHomomorphismByImages">
## <ManSection>
## <Func Name="GroupHomomorphismByImages" Arg='G, H[[, gens], imgs]'/>
##
## <Description>
## <Ref Func="GroupHomomorphismByImages"/> returns the group homomorphism
## with source <A>G</A> and range <A>H</A> that is defined by mapping the
## list <A>gens</A> of generators of <A>G</A> to the list <A>imgs</A> of
## images in <A>H</A>.
## <P/>
## If omitted, the arguments <A>gens</A> and <A>imgs</A> default to
## the <Ref Attr="GeneratorsOfGroup"/> value of <A>G</A> and <A>H</A>,
## respectively. If <A>H</A> is not given the mapping is automatically
## considered as surjective.
## <P/>
## If <A>gens</A> does not generate <A>G</A> or if the mapping of the
## generators does not extend to a homomorphism
## (i.e., if mapping the generators describes only a multi-valued mapping)
## then <K>fail</K> is returned.
## <P/>
## This test can be quite expensive. If one is certain that the mapping of
## the generators extends to a homomorphism,
## one can avoid the checks by calling
## <Ref Oper="GroupHomomorphismByImagesNC"/>.
## (There also is the possibility to
## construct potentially multi-valued mappings with
## <Ref Oper="GroupGeneralMappingByImages"/> and to test with
## <Ref Filt="IsMapping"/> whether they are indeed homomorphisms.)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "GroupHomomorphismByImages" );
#############################################################################
##
#O GroupHomomorphismByImagesNC( <G>, <H>[[, <gens>], <imgs>] )
##
## <#GAPDoc Label="GroupHomomorphismByImagesNC">
## <ManSection>
## <Oper Name="GroupHomomorphismByImagesNC" Arg='G, H[[, gens], imgs]'/>
##
## <Description>
## <Ref Oper="GroupHomomorphismByImagesNC"/> creates a homomorphism as
## <Ref Func="GroupHomomorphismByImages"/> does, however it does not test
## whether <A>gens</A> generates <A>G</A> and that the mapping of
## <A>gens</A> to <A>imgs</A> indeed defines a group homomorphism.
## Because these tests can be expensive it can be substantially faster than
## <Ref Func="GroupHomomorphismByImages"/>.
## Results are unpredictable if the conditions do not hold.
## <P/>
## If omitted, the arguments <A>gens</A> and <A>imgs</A> default to
## the <Ref Attr="GeneratorsOfGroup"/> value of <A>G</A> and <A>H</A>,
## respectively.
## <P/>
## (For creating a possibly multi-valued mapping from <A>G</A> to <A>H</A>
## that respects multiplication and inverses,
## <Ref Oper="GroupGeneralMappingByImages"/> can be used.)
## <!-- If we could guarantee that it does not matter whether we construct the-->
## <!-- homomorphism directly or whether we construct first a general mapping-->
## <!-- and ask it for being a homomorphism,-->
## <!-- then this operation would be obsolete,-->
## <!-- and <C>GroupHomomorphismByImages</C> would be allowed to return the general-->
## <!-- mapping itself after the checks.-->
## <!-- (See also the declarations of <C>AlgebraHomomorphismByImagesNC</C>,-->
## <!-- <C>AlgebraWithOneHomomorphismByImagesNC</C>,-->
## <!-- <C>LeftModuleHomomorphismByImagesNC</C>.)-->
## <P/>
## <Example><![CDATA[
## gap> gens:=[(1,2,3,4),(1,2)];
## [ (1,2,3,4), (1,2) ]
## gap> g:=Group(gens);
## Group([ (1,2,3,4), (1,2) ])
## gap> h:=Group((1,2,3),(1,2));
## Group([ (1,2,3), (1,2) ])
## gap> hom:=GroupHomomorphismByImages(g,h,gens,[(1,2),(1,3)]);
## [ (1,2,3,4), (1,2) ] -> [ (1,2), (1,3) ]
## gap> Image(hom,(1,4));
## (2,3)
## gap> map:=GroupHomomorphismByImages(g,h,gens,[(1,2,3),(1,2)]);
## fail
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "GroupHomomorphismByImagesNC",
[ IsGroup, IsGroup, IsList, IsList ] );
DeclareOperation( "GroupHomomorphismByImagesNC",
[ IsGroup, IsList, IsList ] );
#############################################################################
##
#R IsGroupGeneralMappingByImages(<map>)
##
## <#GAPDoc Label="IsGroupGeneralMappingByImages">
## <ManSection>
## <Filt Name="IsGroupGeneralMappingByImages" Arg='map'
## Type='Representation'/>
##
## <Description>
## Representation for mappings from one group to another that are defined
## by extending a mapping of group generators homomorphically.
## Instead of record components,
## the attribute <Ref Attr="MappingGeneratorsImages"/> is
## used to store generators and their images.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareRepresentation( "IsGroupGeneralMappingByImages",
IsGroupGeneralMapping and IsSPGeneralMapping and IsAttributeStoringRep,
[] );
#############################################################################
##
#R IsPreimagesByAsGroupGeneralMappingByImages(<map>)
##
## <#GAPDoc Label="IsPreimagesByAsGroupGeneralMappingByImages">
## <ManSection>
## <Filt Name="IsPreimagesByAsGroupGeneralMappingByImages" Arg='map'
## Type='Representation'/>
##
## <Description>
## Representation for mappings that delegate work for preimages to a
## mapping created with <Ref Func="GroupHomomorphismByImages"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareRepresentation( "IsPreimagesByAsGroupGeneralMappingByImages",
IsGroupGeneralMapping and IsSPGeneralMapping and IsAttributeStoringRep,
[ ] );
#############################################################################
##
#R IsGroupGeneralMappingByAsGroupGeneralMappingByImages(<map>)
##
## <#GAPDoc Label="IsGroupGeneralMappingByAsGroupGeneralMappingByImages">
## <ManSection>
## <Filt Name="IsGroupGeneralMappingByAsGroupGeneralMappingByImages"
## Arg='map' Type='Representation'/>
##
## <Description>
## Representation for mappings that delegate work on a
## <Ref Func="GroupHomomorphismByImages"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareRepresentation( "IsGroupGeneralMappingByAsGroupGeneralMappingByImages",
IsPreimagesByAsGroupGeneralMappingByImages, [ ] );
#############################################################################
##
#A AsGroupGeneralMappingByImages(<map>)
##
## <#GAPDoc Label="AsGroupGeneralMappingByImages">
## <ManSection>
## <Attr Name="AsGroupGeneralMappingByImages" Arg='map'/>
##
## <Description>
## If <A>map</A> is a mapping from one group to another this attribute
## returns a group general mapping that which implements the same abstract
## mapping. (Some operations can be performed more effective in this
## representation, see
## also <Ref Filt="IsGroupGeneralMappingByAsGroupGeneralMappingByImages"/>.)
## <Example><![CDATA[
## gap> AsGroupGeneralMappingByImages(hom);
## [ (1,2,3,4), (1,2) ] -> [ (1,2), (1,2) ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "AsGroupGeneralMappingByImages", IsGroupGeneralMapping );
#############################################################################
##
#A MappingOfWhichItIsAsGGMBI(<map>)
##
## <ManSection>
## <Attr Name="MappingOfWhichItIsAsGGMBI" Arg='map'/>
##
## <Description>
## If <A>map</A> is <C>AsGroupGeneralMappingByImages(<A>map2</A>)</C> then
## <A>map2</A> is <C>MappingOfWhichItIsAsGGMBI(<A>map</A>)</C>. This attribute is used to
## transfer attribute values which were set later.
## </Description>
## </ManSection>
##
DeclareAttribute( "MappingOfWhichItIsAsGGMBI", IsGroupGeneralMapping );
InstallTrueMethod( IsGroupGeneralMapping, MappingOfWhichItIsAsGGMBI );
InstallAttributeMethodByGroupGeneralMappingByImages :=
function( attr )
InstallMethod( attr, "via `AsGroupGeneralMappingByImages'", true,
[ IsGroupGeneralMappingByAsGroupGeneralMappingByImages ], 0,
hom -> attr( AsGroupGeneralMappingByImages( hom ) ) );
InstallMethod( attr, "get delayed set attribute values", true,
[ IsGroupGeneralMapping and HasMappingOfWhichItIsAsGGMBI ],
SUM_FLAGS-1, # we want to do this before doing any calculations
function(hom)
hom:=MappingOfWhichItIsAsGGMBI( hom );
if Tester(attr)(hom) then
return attr(hom);
else
TryNextMethod();
fi;
end);
end;
#############################################################################
##
## 2. Functions for creating natural homomorphisms
##
#############################################################################
##
#F NaturalHomomorphismByNormalSubgroup( <G>, <N> )
#F NaturalHomomorphismByNormalSubgroupNC( <G>, <N> )
##
## <#GAPDoc Label="NaturalHomomorphismByNormalSubgroup">
## <ManSection>
## <Func Name="NaturalHomomorphismByNormalSubgroup" Arg='G, N'/>
## <Func Name="NaturalHomomorphismByNormalSubgroupNC" Arg='G, N'/>
##
## <Description>
## returns a homomorphism from <A>G</A> to another group whose kernel is <A>N</A>.
## &GAP; will try to select the image group as to make computations in it
## as efficient as possible. As the factor group <M><A>G</A>/<A>N</A></M> can be identified
## with the image of <A>G</A> this permits efficient computations in the factor
## group.
## The homomorphism returned is not necessarily surjective, so
## <Ref Attr="ImagesSource"/> should be used instead of
## <Ref Attr="Range" Label="of a general mapping"/>
## to get a group isomorphic to the factor group.
## The <C>NC</C> variant does not check whether <A>N</A> is normal in
## <A>G</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
InParentFOA( "NaturalHomomorphismByNormalSubgroupNC", IsGroup, IsGroup,
DeclareAttribute );
DeclareSynonym( "NaturalHomomorphismByNormalSubgroupInParent",
NaturalHomomorphismByNormalSubgroupNCInParent );
DeclareSynonym( "NaturalHomomorphismByNormalSubgroupOp",
NaturalHomomorphismByNormalSubgroupNCOp );
#T Get rid of this hack when the ``in parent'' approach is cleaned!
BindGlobal( "NaturalHomomorphismByNormalSubgroupNCOrig",
NaturalHomomorphismByNormalSubgroupNC );
#T Get rid of this hack when the ``in parent'' approach is cleaned!
MakeReadWriteGlobal( "NaturalHomomorphismByNormalSubgroupNC" );
UnbindGlobal( "NaturalHomomorphismByNormalSubgroupNC" );
BindGlobal( "NaturalHomomorphismByNormalSubgroupNC",
function( G, N )
local hom;
hom:= NaturalHomomorphismByNormalSubgroupNCOrig( G, N );
SetIsMapping( hom, true );
return hom;
end );
#T Get rid of this hack when the ``in parent'' approach is cleaned!
DeclareGlobalFunction( "NaturalHomomorphismByNormalSubgroup" );
#############################################################################
##
## 3. Functions for conjugation action
##
#############################################################################
##
#O ConjugatorIsomorphism( <G>, <g> )
##
## <#GAPDoc Label="ConjugatorIsomorphism">
## <ManSection>
## <Oper Name="ConjugatorIsomorphism" Arg='G, g'/>
##
## <Description>
## Let <A>G</A> be a group, and <A>g</A> an element in the same family as
## the elements of <A>G</A>.
## <Ref Oper="ConjugatorIsomorphism"/> returns the isomorphism from <A>G</A>
## to <C><A>G</A>^<A>g</A></C> defined by <M>h \mapsto h^{<A>g</A>}</M>
## for all <M>h \in <A>G</A></M>.
## <P/>
## If <A>g</A> normalizes <A>G</A> then <Ref Oper="ConjugatorIsomorphism"/>
## does the same as <Ref Oper="ConjugatorAutomorphismNC"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "ConjugatorIsomorphism",
[ IsGroup, IsMultiplicativeElementWithInverse ] );
#############################################################################
##
#F ConjugatorAutomorphism( <G>, <g> )
#O ConjugatorAutomorphismNC( <G>, <g> )
##
## <#GAPDoc Label="ConjugatorAutomorphism">
## <ManSection>
## <Func Name="ConjugatorAutomorphism" Arg='G, g'/>
## <Oper Name="ConjugatorAutomorphismNC" Arg='G, g'/>
##
## <Description>
## Let <A>G</A> be a group, and <A>g</A> an element in the same family as
## the elements of <A>G</A> such that <A>g</A> normalizes <A>G</A>.
## <Ref Func="ConjugatorAutomorphism"/> returns the automorphism of <A>G</A>
## defined by <M>h \mapsto h^{<A>g</A>}</M> for all <M>h \in <A>G</A></M>.
## <P/>
## If conjugation by <A>g</A> does <E>not</E> leave <A>G</A> invariant,
## <Ref Func="ConjugatorAutomorphism"/> returns <K>fail</K>;
## in this case,
## the isomorphism from <A>G</A> to <C><A>G</A>^<A>g</A></C> induced by
## conjugation with <A>g</A> can be constructed with
## <Ref Oper="ConjugatorIsomorphism"/>.
## <P/>
## <Ref Oper="ConjugatorAutomorphismNC"/> does the same as
## <Ref Func="ConjugatorAutomorphism"/>,
## except that the check is omitted whether <A>g</A> normalizes <A>G</A>
## and it is assumed that <A>g</A> is chosen to be in <A>G</A> if possible.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ConjugatorAutomorphism" );
DeclareOperation( "ConjugatorAutomorphismNC",
[ IsGroup, IsMultiplicativeElementWithInverse ] );
#############################################################################
##
#F InnerAutomorphism( <G>, <g> )
#O InnerAutomorphismNC( <G>, <g> )
##
## <#GAPDoc Label="InnerAutomorphism">
## <ManSection>
## <Func Name="InnerAutomorphism" Arg='G, g'/>
## <Oper Name="InnerAutomorphismNC" Arg='G, g'/>
##
## <Description>
## Let <A>G</A> be a group, and <M><A>g</A> \in <A>G</A></M>.
## <Ref Func="InnerAutomorphism"/> returns the automorphism of <A>G</A>
## defined by <M>h \mapsto h^{<A>g</A>}</M> for all <M>h \in <A>G</A></M>.
## <P/>
## If <A>g</A> is <E>not</E> an element of <A>G</A>,
## <Ref Func="InnerAutomorphism"/> returns <K>fail</K>;
## in this case,
## the isomorphism from <A>G</A> to <C><A>G</A>^<A>g</A></C> induced by
## conjugation with <A>g</A> can be constructed
## with <Ref Oper="ConjugatorIsomorphism"/>
## or with <Ref Func="ConjugatorAutomorphism"/>.
## <P/>
## <Ref Oper="InnerAutomorphismNC"/> does the same as
## <Ref Func="InnerAutomorphism"/>,
## except that the check is omitted whether <M><A>g</A> \in <A>G</A></M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "InnerAutomorphism" );
DeclareOperation( "InnerAutomorphismNC",
[ IsGroup, IsMultiplicativeElementWithInverse ] );
#############################################################################
##
#P IsConjugatorIsomorphism( <hom> )
#P IsConjugatorAutomorphism( <hom> )
#P IsInnerAutomorphism( <hom> )
##
## <#GAPDoc Label="IsConjugatorIsomorphism">
## <ManSection>
## <Prop Name="IsConjugatorIsomorphism" Arg='hom'/>
## <Prop Name="IsConjugatorAutomorphism" Arg='hom'/>
## <Prop Name="IsInnerAutomorphism" Arg='hom'/>
##
## <Description>
## Let <A>hom</A> be a group general mapping
## (see <Ref Filt="IsGroupGeneralMapping"/>) with source <M>G</M>.
## <Ref Prop="IsConjugatorIsomorphism"/> returns <K>true</K> if <A>hom</A>
## is induced by conjugation of <M>G</M> by an element <M>g</M> that lies in
## <M>G</M> or in a group into which <M>G</M> is naturally embedded
## in the sense described below, and <K>false</K> otherwise.
## <P/>
## Natural embeddings are dealt with in the case that <M>G</M> is
## a permutation group (see Chapter <Ref Chap="Permutation Groups"/>),
## a matrix group (see Chapter <Ref Chap="Matrix Groups"/>),
## a finitely presented group
## (see Chapter <Ref Chap="Finitely Presented Groups"/>), or
## a group given w.r.t. a polycyclic presentation
## (see Chapter <Ref Chap="Pc Groups"/>).
## In all other cases, <Ref Prop="IsConjugatorIsomorphism"/> may return
## <K>false</K> if <A>hom</A> is induced by conjugation
## but is not an inner automorphism.
## <P/>
## If <Ref Prop="IsConjugatorIsomorphism"/> returns <K>true</K> for
## <A>hom</A> then an element <M>g</M> that induces <A>hom</A> can be
## accessed as value of the attribute
## <Ref Attr="ConjugatorOfConjugatorIsomorphism"/>.
## <P/>
## <Ref Prop="IsConjugatorAutomorphism"/> returns <K>true</K> if <A>hom</A>
## is an automorphism (see <Ref Prop="IsEndoGeneralMapping"/>)
## that is regarded as a conjugator isomorphism
## by <Ref Prop="IsConjugatorIsomorphism"/>, and <K>false</K> otherwise.
## <P/>
## <Ref Prop="IsInnerAutomorphism"/> returns <K>true</K> if <A>hom</A> is a
## conjugator automorphism such that an element <M>g</M> inducing <A>hom</A>
## can be chosen in <M>G</M>, and <K>false</K> otherwise.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsConjugatorIsomorphism", IsGroupGeneralMapping );
DeclareSynonymAttr( "IsConjugatorAutomorphism",
IsEndoGeneralMapping and IsConjugatorIsomorphism );
DeclareProperty( "IsInnerAutomorphism", IsGroupGeneralMapping );
InstallTrueMethod( IsBijective, IsConjugatorIsomorphism );
InstallTrueMethod( IsGroupHomomorphism, IsConjugatorIsomorphism );
InstallTrueMethod( IsConjugatorAutomorphism, IsInnerAutomorphism );
#############################################################################
##
#A ConjugatorOfConjugatorIsomorphism( <hom> )
##
## <#GAPDoc Label="ConjugatorOfConjugatorIsomorphism">
## <ManSection>
## <Attr Name="ConjugatorOfConjugatorIsomorphism" Arg='hom'/>
##
## <Description>
## For a conjugator isomorphism <A>hom</A>
## (see <Ref Oper="ConjugatorIsomorphism"/>),
## <Ref Attr="ConjugatorOfConjugatorIsomorphism"/> returns an element
## <M>g</M> such that mapping under <A>hom</A> is induced by conjugation
## with <M>g</M>.
## <P/>
## To avoid problems with <Ref Prop="IsInnerAutomorphism"/>,
## it is guaranteed that the conjugator is taken from the source of
## <A>hom</A> if possible.
## <P/>
## <Example><![CDATA[
## gap> hgens:=[(1,2,3),(1,2,4)];;h:=Group(hgens);;
## gap> hom:=GroupHomomorphismByImages(h,h,hgens,[(1,2,3),(2,3,4)]);;
## gap> IsInnerAutomorphism(hom);
## true
## gap> ConjugatorOfConjugatorIsomorphism(hom);
## (1,2,3)
## gap> hom:=GroupHomomorphismByImages(h,h,hgens,[(1,3,2),(1,4,2)]);
## [ (1,2,3), (1,2,4) ] -> [ (1,3,2), (1,4,2) ]
## gap> IsInnerAutomorphism(hom);
## false
## gap> IsConjugatorAutomorphism(hom);
## true
## gap> ConjugatorOfConjugatorIsomorphism(hom);
## (1,2)
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ConjugatorOfConjugatorIsomorphism",
IsConjugatorIsomorphism );
## just for compatibility with &GAP; 4.1 ...
DeclareSynonymAttr( "ConjugatorInnerAutomorphism",
ConjugatorOfConjugatorIsomorphism );
#############################################################################
##
## 4. Functions for ...
##
DeclareGlobalFunction( "MakeMapping" );
#############################################################################
##
#F GroupHomomorphismByFunction( <S>, <R>, <fun>[, <invfun>] )
#F GroupHomomorphismByFunction( <S>, <R>, <fun>, `false', <prefun> )
##
## <#GAPDoc Label="GroupHomomorphismByFunction">
## <ManSection>
## <Heading>GroupHomomorphismByFunction</Heading>
## <Func Name="GroupHomomorphismByFunction" Arg='S, R, fun[, invfun]'
## Label="by function (and inverse function) between two domains"/>
## <Func Name="GroupHomomorphismByFunction" Arg='S, R, fun, false, prefun'
## Label="by function and function that computes one preimage"/>
##
## <Description>
## <Ref Func="GroupHomomorphismByFunction" Label="by function (and inverse function) between two domains"/>
## returns a group homomorphism
## <C>hom</C> with source <A>S</A> and range <A>R</A>,
## such that each element <C>s</C> of <A>S</A> is mapped to the element
## <A>fun</A><C>( s )</C>, where <A>fun</A> is a &GAP; function.
## <P/>
## If the argument <A>invfun</A> is bound then <A>hom</A> is a bijection
## between <A>S</A> and <A>R</A>,
## and the preimage of each element <C>r</C> of <A>R</A> is given by
## <A>invfun</A><C>( r )</C>,
## where <A>invfun</A> is a &GAP; function.
## <P/>
## If five arguments are given and the fourth argument is <K>false</K> then
## the &GAP; function <A>prefun</A> can be used to compute a single preimage
## also if <C>hom</C> is not bijective.
## <P/>
## No test is performed on whether the functions actually give an
## homomorphism between both groups because this would require testing the
## full multiplication table.
## <P/>
## <Ref Func="GroupHomomorphismByFunction" Label="by function (and inverse function) between two domains"/>
## creates a mapping which lies in <Ref Filt="IsSPGeneralMapping"/>.
## <P/>
## <Example><![CDATA[
## gap> hom:=GroupHomomorphismByFunction(g,h,
## > function(x) if SignPerm(x)=-1 then return (1,2); else return ();fi;end);
## MappingByFunction( Group([ (1,2,3,4), (1,2) ]), Group(
## [ (1,2,3), (1,2) ]), function( x ) ... end )
## gap> ImagesSource(hom);
## Group([ (1,2), (1,2) ])
## gap> Image(hom,(1,2,3,4));
## (1,2)
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("GroupHomomorphismByFunction");
#############################################################################
##
#F ImagesRepresentativeGMBIByElementsList( <hom>, <elm> )
##
## <ManSection>
## <Func Name="ImagesRepresentativeGMBIByElementsList" Arg='hom, elm'/>
##
## <Description>
## This is the method for <C>ImagesRepresentative</C> which calls <C>MakeMapping</C>
## and uses element lists to evaluate the image. It is used by
## <C>Factorization</C>.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("ImagesRepresentativeGMBIByElementsList");
#############################################################################
##
#A ImagesSmallestGenerators(<map>)
##
## <#GAPDoc Label="ImagesSmallestGenerators">
## <ManSection>
## <Attr Name="ImagesSmallestGenerators" Arg='map'/>
##
## <Description>
## returns the list of images of <C>GeneratorsSmallest(Source(<A>map</A>))</C>.
## This list can be used to compare group homomorphisms. (The standard
## comparison is to compare the image lists on the set of elements of the
## source. If however x and y have the same images under a and b,
## certainly all their products have. Therefore it is sufficient to test
## this on the images of the smallest generators.)
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "ImagesSmallestGenerators",
IsGroupGeneralMapping );
#############################################################################
##
#A RegularActionHomomorphism( <G> )
##
## <#GAPDoc Label="RegularActionHomomorphism">
## <ManSection>
## <Attr Name="RegularActionHomomorphism" Arg='G'/>
##
## <Description>
## returns an isomorphism from <A>G</A> onto the regular permutation
## representation of <A>G</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "RegularActionHomomorphism", IsGroup );
DeclareGlobalFunction("IsomorphismAbelianGroupViaIndependentGenerators");
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