The algorithm in Har09 approximates the Schur multiplier of an invariantly finitely L-presented group by the quotients in its Dwyer-filtration. This is implemented in the NQL-package and the following methods are available:
GeneratingSetOfMultiplier(
LpGroup ) A
uses Tietze transformations for computing an equivalent set of relators for LpGroup so that a generating set for its Schur multiplier can be read off easily.
FiniteRankSchurMultiplier(
LpGroup,
c ) O
computes a finitely generated quotient of the Schur multiplier of LpGroup. The method computes the image of the Schur multiplier of LpGroup in the Schur multiplier of its class-c quotient.
EndomorphismsOfFRSchurMultiplier (
LpGroup,
c ) O
computes a list of endomorphisms of the FiniteRankSchurMultiplier
of
LpGroup. These are the endomorphisms of the invariant L-presentation
induced to FiniteRankSchurMultiplier
.
EpimorphismCoveringGroups(
LpGroup,
d,
c ) O
computes an epimorphism of the covering group of the class-d quotient onto the covering group of the class-c quotient.
EpimorphismFiniteRankSchurMultiplier(
LpGroup,
d,
c ) O
computes an epimorphism of the d-th FiniteRankSchurMultiplier
of
the invariant LpGroup onto the c-th FiniteRankSchurMultiplier
.
Its restricts the epimorphism EpimorphismCoveringGroups
to the
corresponding finite rank multipliers.
ImageInFiniteRankSchurMultiplier(
LpGroup,
c,
elm ) F
computes the image of the free group element elm in the c-th
FiniteRankSchurMultiplier
. Note that elm must be a relator contained
in the Schur multiplier of LpGroup; otherwise, the function fails in
computing the image.
vskip3ex
The following example tackels the Schur multiplier of the Grigorchuk group.
gap> G := ExamplesOfLPresentations( 1 );; gap> gens := GeneratingSetOfMultiplier( G ); rec( FixedGens := [ b^-2*c^-2*d^-2*b*c*d*b*c*d ], IteratedGens := [ d^-1*a^-1*d^-1*a*d*a^-1*d*a, d^-1*a^-1*c^-1*a^-1*c^-1*a^-1*d^-1*a*c*a*c*a*d*a^-1*c^-1*a^-1*c^-1*a^ -1*d*a*c*a*c*a ], BasisGens := [ a^2, b*c*d, b^-2*d^-2*b*c*d*b*c*d, b^-2*c^-2*b*c*d*b*c*d ], Endomorphisms := [ [ a, b, c, d ] -> [ a^-1*c*a, d, b, c ] ] ) gap> H := FiniteRankSchurMultiplier( G, 5 ); Pcp-group with orders [ 2, 2, 2 ] gap> GeneratorsOfGroup( H ); [ g15, g17, g16 ] gap> EndomorphismsOfFRSchurMultiplier( G, 5 ); [ [ g15, g16, g17 ] -> [ g15, id, g16 ] ] gap> Kernel( last[1] ); Pcp-group with orders [ 2 ] gap> GeneratorsOfGroup( last ); [ g16 ] gap> EpimorphismFiniteRankSchurMultipliers( G, 5, 2 ); [ g15, g16, g17 ] -> [ g10, id, g13 ] gap> Range( last ) = FiniteRankSchurMultiplier( G, 2 ); true gap> Kernel( EpimorphismFiniteRankSchurMultipliers( G, 5, 2 ) ); Pcp-group with orders [ 2 ] gap> GeneratorsOfGroup( last ); [ g16 ] gap> Kernel( EpimorphismFiniteRankSchurMultipliers( G, 5, 2 ) ) = > Kernel( EndomorphismsOfFRSchurMultiplier( G, 5 )[1] ); true gap> ImageInFiniteRankSchurMultiplier( G, 5, gens.FixedGens[1] ); g15 gap> ImageInFiniteRankSchurMultiplier(G,5,Image(gens.Endomorphisms[1], > gens.IteratedGens[1] ) ); g16 gap> ImageInFiniteRankSchurMultiplier(G,5,gens.IteratedGens[1] ); g17
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