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7. Computing with GAP

7.9: How can one compute the Schur Multiplier of a finite group by GAP?

Derek Holt answered:

One possibility is to use the package cohomolo. 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 (the degree of which is presently restricted to at most 4096). Here is an example.

gap> G:=PSL(3,4);;
gap> chr := CHR(G,2);;
# This constructs a 'cohomology-record' that is used in all further functions
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 gave the warning:"But this package involves external C programs, and it is only possible to use it under Unix or Linux."

However in a further letter Dima Pasechnik added:

In Fact "cohomolo" also runs seemingly OK on Windows with Cygwin (http://www.cygwin.com). Note however that there are a couple of very minor changes in the installation procedure of the package.

Finally Alexander Hulpke added:

Let me just add to Derek Holt's response, that GAP also has a built-in routine for multiplier calcultation. The commands are AbelianInvariantsMultiplier (structure of the multiplier) and EpimorphismSchurCover (map from a covering group, represented as finitely presented group, onto original group)

 
gap> AbelianInvariantsMultiplier(SymmetricGroup(6));    
[ 2 ]    
gap> EpimorphismSchurCover(SymmetricGroup(6)); 
[ f1, f2,  f3, f4, f5 ] -> [(1,2), (2,3), (3,4), (4,5), (5,6)] 

The algorithms used are similar to the ones used by the `cohomolo' package, the main advantage of the library implementation is that it does not require an external program and thus can also run under Windows or if the `cohomolo' package is not installed.

Comparison of runtimes of the two implementations shows that for 'small' groups (say up to order 20000) there is no significant difference, while for large groups the implementation in the cohomolo package runs faster (e.g. in an experiment by Derek Holt for PSp(6,3) of order 4585351680 the corresponding times were 6 and 80 seconds respectively).