This will be an elementary talk defining the first cohomology group H¹(G,M) of a group G acting on a module M, and explaining how to compute it as the nullspace of a system of linear equations. Here M can be an arbitrary finitely generated abelian group.

We describe applications to finding complements in extensions of M by G, and to the computation of subgroups of a finite group and of the automorphism group of a finite group.