On the isomorphism problem for central extensions I

TL;DR

This paper investigates the isomorphism problem for central extensions formed by perturbed direct products of groups, reducing the problem to p-subgroups in certain cases and providing characterizations under specific conditions.

Contribution

It offers new methods to determine isomorphisms of perturbed direct products of groups, especially for abelian torsion and finite groups, simplifying the problem to p-subgroup analysis.

Findings

Isomorphism of perturbed direct products can be decided in specific cases.

Reduction of the problem to p-subgroups for certain group classes.

Characterizations of isomorphisms under various assumptions.

Abstract

Let $G_{2}$ be a group which acts trivially on an abelian group $G_{1}$. As is well known, each perturbed direct product of $G_{1}$ and $G_{2}$ under a 2-cocycle $\varepsilon\in Z^{2}(G_{2},G_{1})$ determines a central extension of $G_{1}$ by $G_{2}$. 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.