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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Werner Nickel, 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
##
#############################################################################
##
#V InfoSchur
##
## <ManSection>
## <InfoClass Name="InfoSchur"/>
##
## <Description>
## </Description>
## </ManSection>
##
DeclareInfoClass( "InfoSchur" );
#############################################################################
##
#O SchurCover(<G>)
##
## <#GAPDoc Label="SchurCover">
## <ManSection>
## <Attr Name="SchurCover" Arg='G'/>
##
## <Description>
## returns one (of possibly several) Schur covers of the group <A>G</A>.
## <P/>
## At the moment this cover is represented as a finitely presented group
## and <Ref Attr="IsomorphismPermGroup"/> would be needed to convert it to a
## permutation group.
## <P/>
## If also the relation to <A>G</A> is needed,
## <Ref Attr="EpimorphismSchurCover"/> should be used.
## <Example><![CDATA[
## gap> g:=Group((1,2,3,4),(1,2));;
## gap> epi:=EpimorphismSchurCover(g);
## [ F1, F2, F3 ] -> [ (1,2), (2,3), (3,4) ]
## gap> Size(Source(epi));
## 48
## gap> f:=FreeGroup("a","b");;
## gap> g:=f/ParseRelators(f,"a2,b3,(ab)5");;
## gap> epi:=EpimorphismSchurCover(g);
## [ a, b ] -> [ a, b ]
## gap> Size(Kernel(epi));
## 2
## ]]></Example>
## <P/>
## If the group becomes bigger, Schur Cover calculations might become
## unfeasible.
## <P/>
## There is another operation, <Ref Attr="AbelianInvariantsMultiplier"/>,
## which only returns the structure of the Schur Multiplier,
## and which should work for larger groups as well.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "SchurCover", IsGroup );
##############################################################################
##
#O EpimorphismSchurCover(<G>[,<pl>])
##
## <#GAPDoc Label="EpimorphismSchurCover">
## <ManSection>
## <Attr Name="EpimorphismSchurCover" Arg='G[, pl]'/>
##
## <Description>
## returns an epimorphism <M>epi</M> from a group <M>D</M> onto <A>G</A>.
## The group <M>D</M> is one (of possibly several) Schur covers of <A>G</A>.
## The group <M>D</M> can be obtained as the <Ref Attr="Source"/> value of
## <A>epi</A>.
## The kernel of <M>epi</M> is the Schur multiplier of <A>G</A>.
## If <A>pl</A> is given as a list of primes,
## only the multiplier part for these primes is realized.
## At the moment, <M>D</M> is represented as a finitely presented group.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "EpimorphismSchurCover", IsGroup );
##############################################################################
##
#A AbelianInvariantsMultiplier(<G>)
##
## <#GAPDoc Label="AbelianInvariantsMultiplier">
## <ManSection>
## <Attr Name="AbelianInvariantsMultiplier" Arg='G'/>
##
## <Description>
## <Index>Multiplier</Index>
## <Index>Schur multiplier</Index>
## returns a list of the abelian invariants of the Schur multiplier of
## <A>G</A>.
## <P/>
## At the moment, this operation will not give any information about how to
## extend the multiplier to a Schur Cover.
## <Example><![CDATA[
## gap> AbelianInvariantsMultiplier(g);
## [ 2 ]
## gap> AbelianInvariantsMultiplier(AlternatingGroup(6));
## [ 2, 3 ]
## gap> AbelianInvariantsMultiplier(SL(2,3));
## [ ]
## gap> AbelianInvariantsMultiplier(SL(3,2));
## [ 2 ]
## gap> AbelianInvariantsMultiplier(PSU(4,2));
## [ 2 ]
## ]]></Example>
## (Note that the last command from the example will take some time.)
## <P/>
## The &GAP; 4.4.12 manual contained examples for larger groups e.g.
## <M>M_{22}</M>. However, some issues that may very rarely (and not
## easily reproducibly) lead to wrong results were discovered in the code
## capable of handling larger groups, and in &GAP; 4.5 it was replaced
## by a more reliable basic method. To deal with larger groups, one can use
## the function <Ref BookName="cohomolo" Func="SchurMultiplier"/> from the
## <Package>cohomolo</Package> package. Also, additional methods for
## <Ref Attr="AbelianInvariantsMultiplier"/> are installed in the
## <Package>Polycyclic</Package> package for pcp-groups.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "AbelianInvariantsMultiplier", IsGroup );
##############################################################################
#### Derived functions. Robert F. Morse
####
##############################################################################
##
#A Epicentre(<G>)
#A ExteriorCentre(<G>)
##
## <#GAPDoc Label="Epicentre">
## <ManSection>
## <Attr Name="Epicentre" Arg='G'/>
## <Attr Name="ExteriorCentre" Arg='G'/>
##
## <Description>
## There are various ways of describing the epicentre of a group <A>G</A>.
## It is the smallest normal subgroup <M>N</M> of <A>G</A> such that
## <M><A>G</A>/N</M> is a central quotient of a group.
## It is also equal to the Exterior Center of <A>G</A>,
## see <Cite Key="Ellis98"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute("Epicentre", IsGroup );
DeclareSynonymAttr("Epicenter", Epicentre);
DeclareSynonymAttr("ExteriorCentre", Epicentre);
DeclareSynonymAttr("ExteriorCenter", Epicentre);
##############################################################################
##
#O NonabelianExteriorSquare(<G>)
##
## <#GAPDoc Label="NonabelianExteriorSquare">
## <ManSection>
## <Oper Name="NonabelianExteriorSquare" Arg='G'/>
##
## <Description>
## Computes the nonabelian exterior square <M><A>G</A> \wedge <A>G</A></M>
## of the group <A>G</A>, which for a finitely presented group is the
## derived subgroup of any Schur cover of <A>G</A>
## (see <Cite Key="BJR87"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation("NonabelianExteriorSquare", [IsGroup]);
##############################################################################
##
#O EpimorphismNonabelianExteriorSquare(<G>)
##
## <#GAPDoc Label="EpimorphismNonabelianExteriorSquare">
## <ManSection>
## <Oper Name="EpimorphismNonabelianExteriorSquare" Arg='G'/>
##
## <Description>
## Computes the mapping
## <M><A>G</A> \wedge <A>G</A> \rightarrow <A>G</A></M>.
## The kernel of this mapping is equal to the Schur multiplier of <A>G</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation("EpimorphismNonabelianExteriorSquare", [IsGroup]);
##############################################################################
##
#P IsCentralFactor(<G>)
##
## <#GAPDoc Label="IsCentralFactor">
## <ManSection>
## <Prop Name="IsCentralFactor" Arg='G'/>
##
## <Description>
## This function determines if there exists a group <M>H</M> such that
## <A>G</A> is isomorphic to the quotient <M>H/Z(H)</M>.
## A group with this property is called in literature <E>capable</E>.
## A group being capable is
## equivalent to the epicentre of <A>G</A> being trivial,
## see <Cite Key="BFS79"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty("IsCentralFactor", IsGroup);
##############################################################################
###########################END RFM############################################
##############################################################################
##
#F SchuMu(<G>,<p>)
##
## <ManSection>
## <Func Name="SchuMu" Arg='G,p'/>
##
## <Description>
## returns epimorphism from p-part of multiplier.p-Sylow (note: This
## extension is <E>not</E> necessarily isomorphic to a Sylow subgroup of a
## Darstellungsgruppe!) onto p-Sylow, the
## kernel is the p-part of the multiplier.
## The implemented algorithm is based on section 7 in Derek Holt's paper.
## However we use some of the general homomorphism setup to avoid having to
## remember certain relations.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("SchuMu");
##############################################################################
##
#F CorestEval(<FG>,<s>)
##
## <ManSection>
## <Func Name="CorestEval" Arg='FG,s'/>
##
## <Description>
## evaluate corestriction mapping.
## <A>FH</A> is a homomorphism from a finitely presented group onto a finite
## group <A>G</A>. <A>s</A> an epimorphism onto a p-Sylow subgroup of <A>G</A> as obtained
## from <C>SchuMu</C>.
## This function evaluates the relators of the source of <A>FH</A> in the
## extension M_p.<A>G</A>. It returns a list whose entries are of the form
## [<A>rel</A>,<A>val</A>], where <A>rel</A> is a relator of <A>G</A> and <A>val</A> its evaluation as
## an element of M_p.
## </Description>
## </ManSection>
##
DeclareGlobalFunction("CorestEval");
##############################################################################
##
#F RelatorFixedMultiplier(<hom>,<p>)
##
## <ManSection>
## <Func Name="RelatorFixedMultiplier" Arg='hom,p'/>
##
## <Description>
## Let <A>hom</A> an epimorphism from an fp group onto a finite group <A>G</A>. This
## function returns an epimorphism onto the <A>p</A>-Sylow subgroup of <A>G</A>,
## whose kernel is the largest quotient of the multiplier, that can lift
## <A>hom</A> to a larger quotient. (The source of this map thus is <M>M_R(B)</M>
## of <Cite Key="HulpkeQuot"/>.)
## </Description>
## </ManSection>
##
DeclareGlobalFunction("RelatorFixedMultiplier");
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