59 Cohomology

This chapter describes functions which may be used to perform certain cohomological calculations on a finite group G.

These include:

:
(i) The p-part Mul_p of the Schur multiplier Mul of G, and a presentation of a covering extension of Mul_p by G, for a specified prime p;
(ii) The dimensions of the first and second cohomology groups of G acting on a finite dimensional KG module M, where K is a field of prime order; and
(iii) Presentations of split and nonsplit extensions of M by G.

All of these functions require G to be defined as a finite permutation group. The functions which compute presentations require, in addition, a presentation of G. Finally, the functions which operate on a module M require the module to be defined by a list of matrices over K. This situation is handled by first defining a {GAP} record, which contains the required information. This is done using the function `CHR`, which must be called before any of the other functions. The remaining functions operate on this record.

If no presentation of the permutation group G is known, and G has order at most 32767, then a presentation can be computed using the function `CalcPres`. On the other hand, if you start with a finitely presented group, then you can create a permutation representation with the function `PermRep` (although there is no guarantee that the representation will be faithful ingeneral).

The functions all compute and make use of a descending sequence of subgroups of G, starting at G and ending with a Sylow p-subgroup of G, and it is usually most efficient to have the indices of the subgroups in this chain as small as possible. If you get a warning message, and one of the function fails because the indices in the chain computed are too large, then you can try to remedy matters by supplying your own chain. See Section Further Information for more details, and an example.

If you set the external variable `InfoCohomolofy` to the value `Print`, then a small amount of information will be printed, indicating what is happening. If chr is the cohomology record you are working with, and you set the field chr`.verbose` to the value `true`, then you will see all the output of the external programs.

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GAP 3.4.4
April 1997