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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Bettina Eick.
##
## 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
##
#############################################################################
##
#I InfoCompPairs
#I InfoExtReps
##
DeclareInfoClass( "InfoCompPairs" );
DeclareInfoClass( "InfoExtReps");
#############################################################################
##
#F MappedPcElement( <elm>, <pcgs>, <list> )
##
## <ManSection>
## <Func Name="MappedPcElement" Arg='elm, pcgs, list'/>
##
## <Description>
## returns the image of <A>elm</A> when mapping the pcgs <A>pcgs</A> onto <A>list</A>
## homomorphically.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("MappedPcElement");
#############################################################################
##
#F TracedPointPcElement( <elm>, <pcgs>, <list>, <pt> )
##
## <ManSection>
## <Func Name="TracedPointPcElement" Arg='elm, pcgs, list,pt'/>
##
## <Description>
## returns the image of <A>pt</A> under the permutation image of <A>elm</A> when mapping the pcgs <A>pcgs</A> onto <A>list</A>
## homomorphically.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("TracedPointPcElement");
#############################################################################
##
#F ExtensionSQ( <C>, <G>, <M>, <c> )
##
## <ManSection>
## <Func Name="ExtensionSQ" Arg='C, G, M, c'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "ExtensionSQ" );
#############################################################################
##
#F FpGroupPcGroupSQ( <G> )
##
## <ManSection>
## <Func Name="FpGroupPcGroupSQ" Arg='G'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "FpGroupPcGroupSQ" );
#############################################################################
##
#F EXPermutationActionPairs( <D> )
##
## <ManSection>
## <Func Name="EXPermutationActionPairs" Arg='D'/>
##
## <Description>
## Let <A>D</A> be a direct product of automorphism group and matrix group as
## used by <C>CompatiblePairs</C>. This function calculates a faithful
## permutation representation of <A>D</A>, which can be used to speed up
## stabilizer calculations. It returns a record with components <C>pairgens</C>:
## Generators of <A>D</A>, <C>permgens</C>: corresponding permutations, <C>permgroup</C>:
## the group generated by <C>permgens</C>, <C>isomorphism</C>: An isomorphism from
## the permutation group to <A>D</A>. This isomorphism can be used to map
## permutations, but it would be hard to get the preimage of a pair in <A>D</A>
## as a permutation.
## The routine may return <K>false</K> if somehow such a representation could
## not be found.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "EXPermutationActionPairs" );
#############################################################################
##
#F EXReducePermutationActionPairs( <r> )
##
## <ManSection>
## <Func Name="EXReducePermutationActionPairs" Arg='r'/>
##
## <Description>
## Let <A>r</A> be a record as returned by <C>EXPermutationActionPairs</C>. This
## function tries to reduce the underlying permutation representation (and
## changes the record accordingly). It is of use when stepping to a
## subgroup of all pairs.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "EXReducePermutationActionPairs" );
#############################################################################
##
#F CompatiblePairs( [<A>,] <G>, <M>[, <D>] )
##
## <#GAPDoc Label="CompatiblePairs">
## <ManSection>
## <Func Name="CompatiblePairs" Arg='[A,] G, M[, D]'/>
##
## <Description>
## returns the group of compatible pairs of the group <A>G</A> with the
## <A>G</A>-module <A>M</A> as subgroup of the direct product
## Aut(<A>G</A>) <M>\times</M> Aut(<A>M</A>).
## Here Aut(<A>M</A>) is considered as subgroup of a general linear group.
## The optional argument <A>D</A> should be a subgroup of
## Aut(<A>G</A>) <M>\times</M> Aut(<A>M</A>).
## If it is given, then only the compatible pairs in <A>D</A> are computed.
## If a group <A>A</A> of automorphisms of <A>G</A> is given as optional
## first argument, it is used in place of the full automorphism group of
## <A>G</A>, avoiding the need to compute this automorphism group.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "CompatiblePairs" );
#############################################################################
##
#O Extension( <G>, <M>, <c> )
#O ExtensionNC( <G>, <M>, <c> )
##
## <#GAPDoc Label="Extension">
## <ManSection>
## <Oper Name="Extension" Arg='G, M, c'/>
## <Oper Name="ExtensionNC" Arg='G, M, c'/>
##
## <Description>
## returns the extension of <A>G</A> by the <A>G</A>-module <A>M</A>
## via the cocycle <A>c</A> as pc groups.
## The <C>NC</C> version does not check the resulting group for consistence.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "Extension", [ CanEasilyComputePcgs, IsObject, IsVector ] );
DeclareOperation( "ExtensionNC", [ CanEasilyComputePcgs, IsObject, IsVector ] );
#############################################################################
##
#O Extensions( <G>, <M> )
##
## <#GAPDoc Label="Extensions">
## <ManSection>
## <Oper Name="Extensions" Arg='G, M'/>
##
## <Description>
## returns all extensions of <A>G</A> by the <A>G</A>-module <A>M</A>
## up to equivalence as pc groups.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "Extensions", [ CanEasilyComputePcgs, IsObject ] );
#############################################################################
##
#O ExtensionRepresentatives( <G>, <M>, <P> )
##
## <#GAPDoc Label="ExtensionRepresentatives">
## <ManSection>
## <Oper Name="ExtensionRepresentatives" Arg='G, M, P'/>
##
## <Description>
## returns all extensions of <A>G</A> by the <A>G</A>-module <A>M</A> up to
## equivalence under action of <A>P</A> where <A>P</A> has to be a subgroup
## of the group of compatible pairs of <A>G</A> with <A>M</A>.
## <Example><![CDATA[
## gap> G := SmallGroup( 4, 2 );;
## gap> mats := List( Pcgs( G ), x -> IdentityMat( 1, GF(2) ) );;
## gap> M := GModuleByMats( mats, GF(2) );;
## gap> A := AutomorphismGroup( G );;
## gap> B := GL( 1, 2 );;
## gap> D := DirectProduct( A, B );; Size(D);
## 6
## gap> P := CompatiblePairs( G, M, D );
## <group of size 6 with 2 generators>
## gap> ExtensionRepresentatives( G, M, P );
## [ <pc group of size 8 with 3 generators>,
## <pc group of size 8 with 3 generators>,
## <pc group of size 8 with 3 generators>,
## <pc group of size 8 with 3 generators> ]
## gap> Extensions( G, M );
## [ <pc group of size 8 with 3 generators>,
## <pc group of size 8 with 3 generators>,
## <pc group of size 8 with 3 generators>,
## <pc group of size 8 with 3 generators>,
## <pc group of size 8 with 3 generators>,
## <pc group of size 8 with 3 generators>,
## <pc group of size 8 with 3 generators>,
## <pc group of size 8 with 3 generators> ]
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "ExtensionRepresentatives",
[CanEasilyComputePcgs, IsObject, IsObject] );
#############################################################################
##
#O SplitExtension( <G>, <M> )
#O SplitExtension( <G>, <aut>, <N> )
##
## <ManSection>
## <Oper Name="SplitExtension" Arg='G, M'/>
## <Oper Name="SplitExtension" Arg='G, aut, N'/>
##
## <Description>
## returns the split extension of <A>G</A> by the <A>G</A>-module <A>M</A>. In the second
## form it returns the split extension of <A>G</A> by the arbitrary finite group
## <A>N</A> where <A>aut</A> is a homomorphism of <A>G</A> into Aut(<A>N</A>).
## </Description>
## </ManSection>
##
DeclareOperation( "SplitExtension", [CanEasilyComputePcgs, IsObject] );
#############################################################################
##
#A SocleComplement(<G>)
##
## <ManSection>
## <Attr Name="SocleComplement" Arg='G'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareAttribute( "SocleComplement", IsGroup );
#############################################################################
##
#A SocleDimensions(<G>)
##
## <ManSection>
## <Attr Name="SocleDimensions" Arg='G'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareAttribute( "SocleDimensions", IsGroup );
#############################################################################
##
#A ModuleOfExtension( < G > );
##
## <ManSection>
## <Attr Name="ModuleOfExtension" Arg='G'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareAttribute( "ModuleOfExtension", IsGroup );
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