Computational Algebra Seminar

Let G = < X | R > be a finitely presented group acting on a KG-module M, with |X|=r, |R|=s, dim(M)=n. In the first talk, we showed how the first cohomology group H¹(G,M) could be computed as

Nullspace(C₁) / Rowspace(C₀),

for a certain n x nr matrix C₀ and nr x ns matrix C₁ over K.

In this talk, we discuss the computation of H²(G,M). This is another important object to compute because the group extensions

0 -> M -> E -> G -> 1

in which the module action of G on M is induced by conjugation in E are classified up to equivalence by H²(G,M).

Once again, the main effort of the computation occurs in the calculation of the nullspace of a certain matrix. This works out nicely when G is a pc-group, but for a general finite group we have to work harder to reduce the size of the matrix involved. If G is a permutation group, then we can sometimes make use of a base and strong generating set to reduce this size to manageable proportions. Furthermore, if we are only interested in the order of the group H²(G,M), rather than in constructing the extensions explicitly, then we can use some cohomology theory to reduce the computation from G to certain p-subgroups of G.