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On the isomorphism problem for central extensions II

Noureddine Snanou

Abstract

In this paper, we study the isomorphism problem for central extensions. More precisely, in some new situations, we provide necessary and sufficient conditions for two central extensions to be isomorphic. We investigate the case when the quotient group is simple or purely non-abelian. Furthermore, we characterize isomorphisms leaving the quotient group invariant. Finally, we deal with isomorphisms of central extensions where the kernel group and the quotient group are isomorphic.

On the isomorphism problem for central extensions II

Abstract

In this paper, we study the isomorphism problem for central extensions. More precisely, in some new situations, we provide necessary and sufficient conditions for two central extensions to be isomorphic. We investigate the case when the quotient group is simple or purely non-abelian. Furthermore, we characterize isomorphisms leaving the quotient group invariant. Finally, we deal with isomorphisms of central extensions where the kernel group and the quotient group are isomorphic.
Paper Structure (6 sections, 18 theorems, 19 equations)

This paper contains 6 sections, 18 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Central extension
  3. Preliminary results
  4. Central extensions with simple or purely non-abelian quotient group
  5. Lower isomorphism problem for central extensions
  6. Isomorphisms of central extension with isomorphic factors group

Key Result

Lemma 3.1

S-C20 Let $\varphi=\left( \right)$ be a group homomorphism from $G_{1}\underset{\varepsilon_{1}}{\times }G_{2}$ to $G_{1}\underset{\varepsilon_{2}}{\times }G_{2}$. Then for all $x\in G_{1}$, and $y\in G_{2}$.

Theorems & Definitions (37)

  • Lemma 3.1
  • Lemma 3.2
  • Definition 3.3
  • Lemma 3.4
  • Proposition 3.5
  • proof
  • Remark 3.6
  • Definition 3.7
  • Proposition 3.8
  • Corollary 3.9
  • ...and 27 more