generalized triangle group
is a group given by a presentation
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
< a, b | ap = bq = Rm = 1 >
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.
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)]);;
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 .
K_iso_Kfp := IsomorphismFpGroup(K);;
gap> Kfp_iso_Kpc:= EpimorphismSolvableQuotient(Kfp,2^20*3^4);;
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[ 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] .|
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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.
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.