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On the isomorphism problem for central extensions I

Noureddine Snanou

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.

On the isomorphism problem for central extensions I

Abstract

Let be a group which acts trivially on an abelian group . As is well known, each perturbed direct product of and under a 2-cocycle determines a central extension of by . 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 -subgroups. We characterize such isomorphisms in various situations with some assumptions on the quotient group.
Paper Structure (5 sections, 13 theorems, 5 equations)

This paper contains 5 sections, 13 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Properties
  3. Isomorphisms inducing the identity on the quotient group
  4. Isomorphisms leaving the kernel group invariant
  5. Isomorphisms inducing a commuting automorphism on the quotient group

Key Result

Proposition 2.1

Let $G_{1}$ be a finite abelian group and $G_{2}$ a finite group and let $\varepsilon \in Z^{2}(G_{2},G_{1})$. If $SZ^{2}(G_{2},G_{1})=\{1\}$, then $G_{1}\underset{\varepsilon}{\times }G_{2}\cong G_{1}\times G_{2}$ if and only if $\varepsilon=1$.

Theorems & Definitions (28)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.4
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Definition 3.4
  • ...and 18 more