Generation of almost simple groups
type: Item
instance of: scholarly article
title: Generation of almost simple groups
zbMATH Open document ID: 0839.20021
author: Francesca Dalla Volta
author: Andrea Lucchini
published in: Journal of Algebra
publication date: 1996-06-10T00:00:00Z
review text: Let \(d(X)\) denote the minimal number of generators for a group \(X\). It is well known that any finite non-abelian simple group is 2-generated. In this paper, the authors prove that if \(G\) is a finite almost simple group, i.e. \(S\leq G\leq \text{Aut} (S)\) for a finite simple non-abelian group \(S\), then \(d(G)= \max\{2, d(G/S)\}\leq 3\) and \(d(G) =3\) if and only if the group \(Z_2\times Z_2\times Z_2\) is an epimorphic image of \(G\) and, in particular, \(S\) is isomorphic to \(A_{2n+1} (p^h)\) or \(D_n (p^h)\), where \(p\) is odd and \(h\) is even. By the same token they give an answer to Gruenberg's problem 11.27 from ``The Kourovka notebook'' [Novosibirsk (1992; Zbl 0831.20003)]. Finally, some consequences about the presentation rank of a finite group are given.
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 826746
zbMATH Keywords: minimal number of generators
zbMATH Keywords: finite non-abelian simple group
zbMATH Keywords: finite almost simple group
zbMATH Keywords: presentation rank
reviewed by: Anatoli Kondrat'ev
MaRDI profile type: Publication
full work available at URL: jabr.1995.1345
OpenAlex ID: W1983764954
DOI: 10.1006/JABR.1995.1345
description: scientific article; zbMATH DE number 826746