The mechanical computation of first and second cohomology groups

We describe the theory and implementation of computer algorithms designed to compute the dimensions of the first and second cohomology groups of a finite group G, acting on a finite module M defined over a field K of prime order. Presentations of extensions of M by G can also be computed. The method is to find a Sylow p-subgroup P of G, where p =❘K❘, to compute Hx (P, M) first, using variants of the Nilpotent Quotient Algorithm, and then to compute Hx (G, M) as the subgroup of stable elements of Hx (P, M)