[GAP Forum] Schur multiplier of GL(3,4)?

[Ellis, Grahamj]

Dear Hung,

You could also compute the Schur multiplier (= second integral homology) of GL(3,4) using the HAP package. As computations tend to be quicker on permutations groups than on matrix groups, use the following commands.

gap> G:=Image(IsomorphismPermGroup(GL(3,4)));;
gap> GroupHomology(G,2);
[  ]

All the best,

Graham 

School of Mathematics, Statistics and Applied Mathematics 
National University of Ireland, Galway
http://hamilton.nuigalway.ie



-----Original Message-----
From: forum-bounces at gap-system.org on behalf of Derek Holt
Sent: Sat 19/03/2011 18:49
To: Nguyen,Hung Ngoc
Cc: forum at gap-system.org
Subject: Re: [GAP Forum] Schur multiplier of GL(3,4)?
 

Dera Hung, Dear Forum,

Yes, you can do it with cohomolo. cohomolo works only with permutation groups,
and you have to do the calculation one prime at at a time.

gap> LoadPackage("cohomolo");
true
gap> G:=GL(3,4);;
gap> P:=Image(IsomorphismPermGroup(G));
<permutation group of size 181440 with 2 generators>
gap> C:=CHR(P,2);; SchurMultiplier(C);
[  ]
gap> C:=CHR(P,3);; SchurMultiplier(C);
[  ]
gap> C:=CHR(P,5);; SchurMultiplier(C);
The Sylow p-subgroup of the group is cyclic - so the multiplier is trivial.
[  ]
gap> C:=CHR(P,7);; SchurMultiplier(C);        
The Sylow p-subgroup of the group is cyclic - so the multiplier is trivial.
[  ]

So the Schur multiplier of GL(3,4) is trivial.

(SL(3,4) on the other hand has multiplier 4 X 4, but the outer automorphism
of SL(3,4) of order 3 induces a fixed-point-free action on this group, and
hence kills it in GL(3,4).)

Derek.

> Dear Forum,
> 
> I am trying to compute the Schur multiplier of GL(3,4) and here is what I got:
> 
> >G:=GeneralLinearGroup(3,4);
> [GL(3,4)
> >AbelianInvariantsMultiplier(G);
> [Error, the coset enumeration has defined more than 256000 cosets
> 
> I just wonder if we can do it by using the ''cohomolo'' package and  working under UNIX/LINUX systems? 
> 
> Thank you very much for your help,
> Hung.
> _______________________________________________
> Forum mailing list
> Forum at mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum

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