Dear GAP Forum, Deepak Naidu wrote:
> Let H be a normal subgroup of a finite group G. The action of G on H, by conjugation, > induces an action of G on the Schur multiplier H^2(H, C*) of H. > Is there any way I can use GAP to find the number of orbits of H^2(H, C*) under the action of G? The following commands (which use the HAP package) show that the there are 4 orbits in the Schur multiplier of N:=AltenratingGroup(6) under the conjugation action of G:=SymmetricGroup(6). Note that the Schur multiplier of N is isomorphic to the Second integral homology of N, and that N acts trivial on its Schur multiplier/second homology (for any group N). So we are really interested in orbits under an action of the quotient group G/N. ############## CUT ###################### gap> G:=SymmetricGroup(6);; gap> N:=AlternatingGroup(6);; gap> R:=ResolutionFiniteGroup(N,3);; gap> SchurMultiplier:=Homology(TensorWithIntegers(R),2); [6] gap> #So the Schur multiplier of N is cyclic of order 6. gap> ConjugationHomomorphism:=function(g); > return GroupHomomorphismByFunction(N,N,x->g*x*g^-1); end;; gap> HomologyMapInducedByConjugation:=function(g) > local f; > f:=EquivariantChainMap(R,R,ConjugationHomomorphism(g)); > f:=TensorWithIntegers(f); > f:=Homology(f,2); > return f; > end;; gap> #Note that (1,2) represents a nontrivial element in G/N. gap> HM:=HomologyMapInducedByConjugation((1,2)); [ f1, f2, f3, f4, f5, f6 ] -> [ f1^4*f2^-3, f1^2*f2^-1, f3, f3^-2*f4^2*f5, f3^-2*f4^2*f5, f6 ] gap> IsomorphismPermGroup(Source(HM)); [ f1, f2, f3, f4, f5, f6 ] -> [ (), (1,2,3)(4,5,6), (1,4)(2,5)(3,6), (), (), ((1,4)((2,5)(3,6) ] ##################### CUT ################## Here Source(HM) is a finitely presented group isomorphic to the cyclic group H_2(N) of order 6.We see that conjugation by (1,2) in G induces an endomorphism on H_2(N)=C_2 x C_3 which fixes the generator f3 of order two, and inverts the generator f2 of order three. Graham _______________________________________________ Forum mailing list [email protected] http://mail.gap-system.org/mailman/listinfo/forum
