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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
##
#############################################################################
##
#F CollectedWordSQ( <C>, <u>, <v> )
##
## <ManSection>
## <Func Name="CollectedWordSQ" Arg='C, u, v'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "CollectedWordSQ" );
#############################################################################
##
#F CollectorSQ( <G>, <M>, <isSplit> )
##
## <ManSection>
## <Func Name="CollectorSQ" Arg='G, M, isSplit'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "CollectorSQ" );
#############################################################################
##
#F AddEquationsSQ( <eq>, <t1>, <t2> )
##
## <ManSection>
## <Func Name="AddEquationsSQ" Arg='eq, t1, t2'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "AddEquationsSQ" );
#############################################################################
##
#F SolutionSQ( <C>, <eq> )
##
## <ManSection>
## <Func Name="SolutionSQ" Arg='C, eq'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "SolutionSQ" );
#############################################################################
##
#F TwoCocyclesSQ( <C>, <G>, <M> )
##
## <ManSection>
## <Func Name="TwoCocyclesSQ" Arg='C, G, M'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "TwoCocyclesSQ" );
#############################################################################
##
#F TwoCoboundariesSQ( <C>, <G>, <M> )
##
## <ManSection>
## <Func Name="TwoCoboundariesSQ" Arg='C, G, M'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "TwoCoboundariesSQ" );
#############################################################################
##
#F TwoCohomologySQ( <C>, <G>, <M> )
##
## <ManSection>
## <Func Name="TwoCohomologySQ" Arg='C, G, M'/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "TwoCohomologySQ" );
#############################################################################
##
#O TwoCocycles( <G>, <M> )
##
## <#GAPDoc Label="TwoCocycles">
## <ManSection>
## <Oper Name="TwoCocycles" Arg='G, M'/>
##
## <Description>
## returns the <M>2</M>-cocycles of a pc group <A>G</A> by the
## <A>G</A>-module <A>M</A>.
## The generators of <A>M</A> must correspond to the <Ref Attr="Pcgs"/>
## value of <A>G</A>. The operation
## returns a list of vectors over the field underlying <A>M</A> and the
## additive group generated by these vectors is the group of
## <M>2</M>-cocyles.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "TwoCocycles", [ IsPcGroup, IsObject ] );
#############################################################################
##
#O TwoCoboundaries( <G>, <M> )
##
## <#GAPDoc Label="TwoCoboundaries">
## <ManSection>
## <Oper Name="TwoCoboundaries" Arg='G, M'/>
##
## <Description>
## returns the group of <M>2</M>-coboundaries of a pc group <A>G</A> by the
## <A>G</A>-module <A>M</A>.
## The generators of <A>M</A> must correspond to the <Ref Attr="Pcgs"/>
## value of <A>G</A>.
## The group of coboundaries is given as vector space over the field
## underlying <A>M</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "TwoCoboundaries", [ IsPcGroup, IsObject ] );
#############################################################################
##
#O TwoCohomology( <G>, <M> )
##
## <#GAPDoc Label="TwoCohomology">
## <ManSection>
## <Oper Name="TwoCohomology" Arg='G, M'/>
##
## <Description>
## This operation computes the second cohomology group for the special
## case of a Pc Group.
## It returns a record defining the second cohomology group as factor space of
## the space of cocycles by the space of coboundaries.
## <A>G</A> must be a pc group and the generators of <A>M</A> must
## correspond to the pcgs of <A>G</A>.
## <Example><![CDATA[
## gap> G := SmallGroup( 4, 2 );
## <pc group of size 4 with 2 generators>
## gap> mats := List( Pcgs( G ), x -> IdentityMat( 1, GF(2) ) );
## [ [ <a GF2 vector of length 1> ], [ <a GF2 vector of length 1> ] ]
## gap> M := GModuleByMats( mats, GF(2) );
## rec( IsOverFiniteField := true, dimension := 1, field := GF(2),
## generators := [ <an immutable 1x1 matrix over GF2>,
## <an immutable 1x1 matrix over GF2> ], isMTXModule := true )
## gap> TwoCoboundaries( G, M );
## [ ]
## gap> TwoCocycles( G, M );
## [ [ Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), Z(2)^0, 0*Z(2) ],
## [ 0*Z(2), 0*Z(2), Z(2)^0 ] ]
## gap> cc := TwoCohomology( G, M );;
## gap> cc.cohom;
## <linear mapping by matrix, <vector space of dimension 3 over GF(
## 2)> -> ( GF(2)^3 )>
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "TwoCohomology", [ IsPcGroup, IsObject ] );
#############################################################################
##
#O TwoCohomologyGeneric( <G>, <M> )
##
## <#GAPDoc Label="TwoCohomologyGeneric">
## <ManSection>
## <Oper Name="TwoCohomologyGeneric" Arg='G, M'/>
##
## <Description>
## This function computes the second cohomology group for an arbitrary
## finite group <A>G</A>. The generators of the module <A>M</A> must
## correspond to the generators of <A>G</A>.
##
## It returns a record with components <C>coboundaries</C>,
## <C>cocycles</C> and <C>cohomology</C> which are lists of vectors that form a
## basis of the respective group. <C>cohomology</C> is chosen as a vector space
## complement to <C>coboundaries</C> in the <C>cocycles</C>.
## These vectors are representing tails in <A>M</A> with respect to the
## <C>relators</C> of the presentation <C>presentation</C> of <A>G</A>.
## (Note that this presentation is on a generating set chosen by the
## routine, this generating system corresponds to the components
## <C>group</C> and <C>module</C> of the record returned.
## The extension corresponding to a cocyle <C>c</C> can be constructed as
## <C>FpGroupCocycle(r,c)</C> where <C>r</C> is the cohomology record. This is
## currently done as a finitely presented group.
##
## <Example><![CDATA[
## gap> g:=Group((1,2,3,4,5),(1,2,3));;
## gap> mats:=[[[2,0,0,1],[1,2,1,0],[2,1,1,1],[2,1,1,0]],
## > [[0,2,0,0],[1,2,1,0],[0,0,1,0],[0,0,0,1]]]*Z(3)^0;;
## gap> mo:=GModuleByMats(mats,GF(3));;
## gap> coh:=TwoCohomologyGeneric(g,mo);;
## gap> coh.cocycles;
## < immutable compressed matrix 8x44 over GF(3) >
## gap> coh.coboundaries;
## [ < immutable compressed vector length 44 over GF(3) >,
## < immutable compressed vector length 44 over GF(3) >,
## < immutable compressed vector length 44 over GF(3) >,
## < immutable compressed vector length 44 over GF(3) >,
## < immutable compressed vector length 44 over GF(3) >,
## < immutable compressed vector length 44 over GF(3) >,
## < immutable compressed vector length 44 over GF(3) > ]
## gap> coh.cohomology;
## [ < immutable compressed vector length 44 over GF(3) > ]
## gap> g1:=FpGroupCocycle(coh,coh.zero,true);
## <fp group of size 4860 on the generators [ F1, F2, F3, m1, m2, m3, m4 ]>
## gap> g2:=FpGroupCocycle(coh,coh.cohomology[1],true);
## <fp group of size 4860 on the generators [ F1, F2, F3, m1, m2, m3, m4 ]>
## gap> g1:=Image(IsomorphismPermGroup(g1));
## <permutation group with 7 generators>
## gap> Length(ComplementClassesRepresentatives(g1,SolvableRadical(g1)));
## 3
## gap> g2:=Image(IsomorphismPermGroup(g2));
## <permutation group with 7 generators>
## gap> Length(ComplementClassesRepresentatives(g2,SolvableRadical(g2)));
## 0
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "TwoCohomologyGeneric", [ IsGroup, IsObject ] );
#############################################################################
##
#O FpGroupCocycle( <r>, <c>, <doperm> )
##
## <#GAPDoc Label="FpGroupCocycle">
## <ManSection>
## <Func Name="FpGroupCocycle" Arg='r, c [,doperm]'/>
##
## <Description>
## For a record <A>r</A> as returned by
## <Ref Oper="TwoCohomologyGeneric"/> and a vector <A>c</A> in the space of
## two-cocycles, this operation returns a finitely presented group that is
## an extension corresponding to the cocycle <A>c</A>. If the underlying
## module has dimension <M>d</M>, the last <M>d</M> generators generate the
## normal subgroup.
## If the optional parameter <A>doperm</A> is given as <A>true</A>, a
## faithful permutation representation is computed and stored in the
## attribute <Ref Attr="IsomorphismPermGroup"/> of the computed group.
## If the option <C>normalform</C> is given as <C>true</C>, arithmetic in
## the resulting finitely presented group will bring words into normal form.
## <Example><![CDATA[
## gap> g:=Group((2,15,8,16)(3,17,14,21)(4,23,20,6)(5,9,22,11)(7,13,19,25),
## > (2,12,7,17)(3,18,13,23)(4,24,19,9)(5,10,25,15)(6,11,16,21));;
## gap> StructureDescription(g);
## "GL(2,5)"
## gap> mats:=[[[1,1,0,2],[2,0,0,0],[0,2,2,0],[0,1,0,0]],
## > [[0,0,0,1],[1,1,2,0],[1,0,2,1],[1,0,1,0]]]*Z(3)^0;;
## gap> mo:=GModuleByMats(mats,GF(3));;
## gap> coh:=TwoCohomologyGeneric(g,mo);;
## gap> coh.cohomology;
## [ < immutable compressed vector length 116 over GF(3) > ]
## gap> p:=FpGroupCocycle(coh,coh.zero,true);
## <fp group of size 38880 on the generators
## [ F1, F2, F3, F4, F5, F6, m1, m2, m3, m4 ]>
## gap> g1:=Image(IsomorphismPermGroup(p));
## <permutation group with 10 generators>
## gap> p:=FpGroupCocycle(coh,coh.cohomology[1],true);
## <fp group of size 38880 on the generators
## [ F1, F2, F3, F4, F5, F6, m1, m2, m3, m4 ]>
## gap> g2:=Image(IsomorphismPermGroup(p));
## <permutation group with 10 generators>
## gap> Collected(List(MaximalSubgroupClassReps(g1),Size));
## [ [ 480, 3 ], [ 3888, 1 ], [ 6480, 1 ], [ 7776, 1 ], [ 19440, 1 ] ]
## gap> Collected(List(MaximalSubgroupClassReps(g2),Size));
## [ [ 3888, 1 ], [ 6480, 1 ], [ 7776, 1 ], [ 19440, 1 ] ]
## gap> p:=FpGroupCocycle(coh,coh.cohomology[1],true:normalform);;
## ]]></Example>
## <Log><![CDATA[
## gap> p.7*p.1; # i.e. m1*F1, but in normal form
## F1*m4
## ]]></Log>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("FpGroupCocycle");
#############################################################################
##
#O CompatiblePairOrbitRepsGeneric( <compair>,<coh> )
##
## <#GAPDoc Label="CompatiblePairOrbitRepsGeneric">
## <ManSection>
## <Func Name="CompatiblePairOrbitRepsGeneric" Arg='compair, coh'/>
##
## <Description>
## <Example><![CDATA[
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction("CompatiblePairOrbitRepsGeneric");
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