[GAP Forum] Schur Multiplier

[Derek Holt]

On Mon, Nov 15, 2004 at 10:31:22PM -0800, Ahmad Erfanian wrote:
> 
> Dear users,
> 
> I would like to ask that  how we can compute the Schur Multiplier of a given finite group by Gap.
> 
> With thanks,
> 
> A. Erfanian.

One possibility is to use the package "cohomolo". But this package involves
external C programs, and it is only possible to use it under Unix or Linux.
You first have to follow the instructions in the package directory to
compile the external programs.

It calculates the Schur multiplier with respect to a given prime, so you need
to try all primes dividing the group order. The input group must be a
permutation group. Here is an example.

gap> G:=PSL(3,4);;
gap> chr := CHR(G,2);;
gap> SchurMultiplier(chr);
[ 4, 4 ]
gap> chr := CHR(G,3);;    
gap> SchurMultiplier(chr);
[ 3 ]
gap> chr := CHR(G,5);;    
gap> SchurMultiplier(chr);
The Sylow p-subgroup of the group is cyclic - so the multiplier is trivial.
[  ]
gap> chr := CHR(G,7);;    
gap> SchurMultiplier(chr);
The Sylow p-subgroup of the group is cyclic - so the multiplier is trivial.
[  ]

We conclude that the Schur multiplier is a direct product of cyclic groups
of order 4, 4 and 3.

It is also possible to compute finite presentations of covering groups.

Derek Holt.