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 <td style="vertical-align: top;"><a
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 <td
style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big>About HAP: The
Rosenberger monster - an example of how to piece together information 
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href="aboutCocycles.html">next</a> 
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 </ 82,106,149, ,158 ,109,151 ,,94 ,, ,90
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style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: left;">A
 generalized triangle group
is a group given by a presentation 
 <div style="text-align: center;"> 
< a, b | ap = bq = Rm = 1 > 
 
 </div>
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 
 <ul>
 <li>[J. Howie, V.Metaftsis & R.M.Thomas, Triangle groups
and their generalizations, Groups
Korea '94, pages 135-147, (de Gruyter 1995)], </li>
 <li>[L. Levai, G. Rosenberger and B. Souvignier, All finite
generalized triangle , span style="font-java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 75
American Mathematical Society 347 (9) (1995), 3625-3627].</li>
 </ul>
The largest finite generalized triangle group is 
 
 <div style="text-align: center;">G = < a, b
| a2, b3, (abababab2ab2abab2ab2)2
> 
 </div>
 
and has order |G| = 220 34 5 . It has been named
the Rosenberger monster. The
following R. a
solvable subgroup K in G of index 5. 
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style="background-color: rgb(255, 255, 204); vertical-align: top;">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 
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style="background-color: rgb(255, 255, 255); vertical-align: top;">The
Schur Multiplier H2(G,Z) of the Rosenberger monster is a
quotient of H2(K,Z) since K contains 118, 147,169 , 167,112,117 144,, 166,90,165,98 ,161
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 . 
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style="background-color: rgb(255, 255, 204); vertical-align: top;">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( 141,112, 110 124 168,162,,90128 162 ,114 ,
 
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 ] 
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
following commands construct the group D=[[G,G],[G,G]] and show that H2(G/D,Z)=Z6
and D=[G,D] .</td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 204);">gap>
D:=DerivedSubgroup(DerivedSubgroup(G));; 
gap> GroupHomology(G/D,2); 
[ 2, 3 ] 
gap> Index(G,D)=Index(G,CommutatorSubgroup(G,D)); 
true 
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">The
five term exact sequence in integral homology (see <a
href="aboutSchurMultiplier.html">this page</a> for details) arising
from the normal subgroup D in G yields a surjection H2(G,Z)
→ H(G/,).We concludethattheSchurmultiplier of the
Rosenberger monster is either 
 
 <div style="text-align: center;">H2(G,Z) = Z3+Z6
 or 
H2(G,Z) = Z6 .. 
 <div style="text-align: left;"> 
We can complete the calculation of H2(G,Z) using the
following basic result. 
 
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style="vertical-align: top; background-color: rgb(204, 255, 255);">For
any finite group G with presentation G = < x | 2(G,Z) satisfies 
 
 <div style="text-align: center;">dH2(G,Z) < |r| - |x| + 1 . 
 </div>
 </td>
 </tr>
 </tbody>
 </table>
 
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. 
 
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 The Rosenberger monster has Schur multiplier H2(G,Z)
= Z6 . 
 </td>
 </tr>
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 </table>
 
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