On the isomorphism problem for central extensions I

Let G₂ be a group which acts trivially on an abelian group G₁. As is well known, each perturbed direct product of G₁ and G₂ under a 2-cocycle ε∈ Z²(G₂,G₁) determines a central extension of G₁ by G₂. The purpose of this paper is to study perturbed direct products of groups and to decide in some cases how the isomorphism of these groups can be decided. Furthermore, we show that the study of the isomorphism of perturbed direct products of an abelian torsion group and a finite group is reduced to the study of the isomorphism of p-subgroups. We characterize such isomorphisms in various situations with some assumptions on the quotient group.