AboutHap

About HAP: The Rosenberger monster - an example of how to piece together information

A generalized triangle group is a group given by a presentation

< a, b | a^p = b^q = R^m = 1 >

where p, q , m are integers greater than 1 and R is a reduced word in the free monoid on a and b. The finite generalized triangle groups were classified in

[J. Howie, V.Metaftsis & R.M.Thomas, Triangle groups and their generalizations, Groups Korea '94, pages 135-147, (de Gruyter 1995)],
[L. Levai, G. Rosenberger and B. Souvignier, All finite generalized triangle groups, Transactions American Mathematical Society 347 (9) (1995), 3625-3627].

The largest finite generalized triangle group is

G = < a, b | a², b³, (abababab²ab²abab²ab²)² >

and has order |G| = 2²⁰ 3⁴ 5 . It has been named the Rosenberger monster. The following GAP commands, which were shown to me by R.F. Morse, create a solvable subgroup K in G of index 5.

gap> F := FreeGroup("a","b");; a:=F.1;; b:=F.2;;
gap> R := [a^2, b^3, (a*b*a*b*a*b*a*b^2*a*b^2*a*b*a*b^2*a*b^2)^2];;
gap> G := F/R;;

gap> a:= G.1;; b:=G.2;;
gap> K:=Subgroup(G, [a, (b^a)^b, (b^a)^(b^2)]);;
gap> Index(G,K);
5

The Schur Multiplier H₂(G,Z) of the Rosenberger monster is a quotient of H₂(K,Z) since K contains the Sylow 2-subgroup and Sylow 3-subgroup of G, and the Sylow 5-subgroup is cyclic and thus has trivial Schur multiplier. The following commands first create a pc-group Kpc isomorphism to K, then construct three terms of a resolution for K, and finally show that H₁(K,Z)=H₂(K,Z) = Z₃+Z₆ .

gap> K_iso_Kfp := IsomorphismFpGroup(K);;
gap> Kfp:=Image(K_iso_Kfp);;
gap> Kfp_iso_Kpc:= EpimorphismSolvableQuotient(Kfp,2^20*3^4);;
gap> Kpc:=Image(Kfp_iso_Kpc);;

gap> D:=DerivedSeries(Kpc)[3];;
gap> NatHom:=NaturalHomomorphismByNormalSubgroup(Kpc,D);;
gap> Q:=Image(NatHom);;

gap> N:=NormalSubgroups(Q)[12];;
gap> RQ:=ResolutionNormalSeries([Q,N,TrivialSubgroup(Q)],3);;

gap> RD:=ResolutionNilpotentGroup(D,3);;
gap> RKpc:=ResolutionExtension(NatHom,RD,RQ);;

gap> TRKpc:=TensorWithIntegers(RKpc);;
gap> Homology(TRKpc,1);
[ 3, 6 ]

gap> Homology(TRKpc,2);
[ 3, 6 ]

The following commands construct the group D=[[G,G],[G,G]] and show that H₂(G/D,Z)=Z₆ and D=[G,D] .

gap> D:=DerivedSubgroup(DerivedSubgroup(G));;
gap> GroupHomology(G/D,2);
[ 2, 3 ]
gap> Index(G,D)=Index(G,CommutatorSubgroup(G,D));
true

The five term exact sequence in integral homology (see this page for details) arising from the normal subgroup D in G yields a surjection H₂(G,Z) → H₂(G/D,Z) . We conclude that the Schur multiplier of the Rosenberger monster is either

H₂(G,Z) = Z₃+Z₆ or H₂(G,Z) = Z₆._.

We can complete the calculation of H₂(G,Z) using the following basic result.

For any finite group G with presentation G = < x | r > the minimum number of generator of the Schur multiplier dH₂(G,Z) satisfies

dH₂(G,Z) < |r| - |x| + 1 .

This result follows from the fact that H_*(G,Z) is finite and is the homology of a chain complex of free abelian groups with 1 generator in dimension 0, |x| generators in dimension 1 and |r| generators in dimension 2. We now have the following.

Proposition

The Rosenberger monster has Schur multiplier H₂(G,Z) = Z₆ .

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