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#############################################################################
##
#W schur.gd GAP library Werner Nickel
#W Alexander Hulpke
##
#Y (C) 2000 School Math and Comp. Sci., University of St. Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
Revision.schur_gd :=
"@(#)$Id: schur.gd,v 4.8.4.2 2006/08/17 14:41:21 gap Exp $";
##############################################################################
##
#V InfoSchur()
##
DeclareInfoClass( "InfoSchur" );
##############################################################################
##
#O SchurCover(<G>)
##
## returns one (of possibly several) Schur covers of <G>.
##
## At the moment this cover is represented as a finitely presented group
## and `IsomorphismPermGroup' would be needed to convert it to a
## permutation group.
##
## If also the relation to <G> is needed, `EpimorphismSchurCover' should be
## used.
##
DeclareAttribute( "SchurCover", IsGroup );
##############################################################################
##
#O EpimorphismSchurCover(<G>[,<pl>])
##
## returns an epimorphism <epi> from a group <D> onto <G>. The group <D> is
## one (of possibly several) Schur covers of <G>.
## The group <D> can be obtained as the `Source' of <epi>. the kernel of
## <epi> is the schur multiplier of <G>.
## If <pl> is given as a list of primes, only the multiplier part for these
## primes is realized.
## At the moment, <D> is represented as a finitely presented group.
DeclareAttribute( "EpimorphismSchurCover", IsGroup );
##############################################################################
##
#A AbelianInvariantsMultiplier(<G>)
##
## \index{Multiplier}\atindex{Schur multiplier}{@Schur multiplier}
## returns a list of the abelian invariants of the Schur multiplier of <G>.
DeclareAttribute( "AbelianInvariantsMultiplier", IsGroup );
##############################################################################
#### Derived functions. Robert F. Morse
####
##############################################################################
##
#A Epicentre(<G>)
#A ExteriorCentre(<G>)
##
## There are various ways of describing the epicentre of a group. It is
## the smallest normal subgroup $N$ of $G$ such that $G/N$ is a central
## quotient of a group. It is also equal to the Exterior Center of $G$
## \cite{Ellis98}.
##
DeclareAttribute("Epicentre", IsGroup );
DeclareSynonym("Epicenter", Epicentre);
DeclareSynonym("ExteriorCentre", Epicentre);
DeclareSynonym("ExteriorCenter", Epicentre);
##############################################################################
##
#O NonabelianExteriorSquare(<G>)
##
## Computes the Nonabelian Exterior Square $G\wedge G$ of a group $G$
## which for a finitely presented group is the derived subgroup of
## any Schur Cover of $G$ \cite{BJR87}.
##
DeclareOperation("NonabelianExteriorSquare", [IsGroup]);
##############################################################################
##
#O EpimorphismNonabelianExteriorSquare(<G>)
##
## Computes the mapping $G\wedge G \to G$. The kernel of this
## mapping is equal to the Schur Multiplicator of $G$.
##
DeclareOperation("EpimorphismNonabelianExteriorSquare", [IsGroup]);
##############################################################################
##
#P IsCentralFactor(<G>)
##
## This method
## determines if there exists a group $H$ such that <G> is isomormorphic
## to the quotient $H/Z(H)$. A group with this property is called in
## literature *capable*.
## A group being capable is
## equivalent to the Epicentre of $G$ being trivial \cite{BFS79}.
##
DeclareProperty("IsCentralFactor", IsGroup);
##############################################################################
###########################END RFM############################################
##############################################################################
##
#F SchuMu(<G>,<p>)
##
## returns epimorphism from p-part of multiplier.p-Sylow (note: This
## extension is *not* 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.
DeclareGlobalFunction("SchuMu");
##############################################################################
##
#F CorestEval(<FG>,<s>)
##
## evaluate corestriction mapping.
## <FH> is an homomorphism from a finitely presented group onto a finite
## group <G>. <s> an epimorphism onto a p-Sylow subgroup of <G> as obtained
## from `SchuMu'.
## This function evaluates the relators of the source of <FH> in the
## extension M_p.<G>. It returns a list whose entries are of the form
## [<rel>,<val>], where <rel> is a relator of <G> and <val> its evaluation as
## an element of M_p.
DeclareGlobalFunction("CorestEval");
##############################################################################
##
#F RelatorFixedMultiplier(<hom>,<p>)
##
## Let <hom> an epimorphism from an fp group onto a finite group <G>. This
## function returns an epimorphism onto the <p>-Sylow subgroup of <G>,
## whose kernel is the largest quotient of the multiplier, that can lift
## <hom> to a larger quotient. (The source of this map thus is $M_R(B)$
## of~\cite{HulpkeQuot}.)
DeclareGlobalFunction("RelatorFixedMultiplier");
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