A finitary Tits' alternative
type: Item
instance of: scholarly article
title: A finitary Tits' alternative
zbMATH Open document ID: 0942.20007
author: Sarah Black
published in: Archiv der Mathematik
publication date: 2000-05-28T00:00:00Z
review text: For a group \(G\), let \(d(G)\) denote the minimal number of generators, and define the rank of~\(G\) to be \(\text{rk}(G)=\sup\{d(H)\mid H\) a subgroup of \(G\}\). In this note, the author proves a version of the well-known Tits alternative that applies to finite groups. Namely, given any family of finite groups of uniformly bounded rank, then either a subdirect product of these groups contains a non-cyclic free group, or each group contains a subgroup of bounded index whose derived subgroup is nilpotent of bounded class. This is based on a similar result dealing with a single finite group, where the first alternative is replaced by the existence of a finite approximation to the free group.
reviewed by: Michael Stoll
Mathematics Subject Classification ID: 20D99
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1295478
zbMATH Keywords: Tits alternative
zbMATH Keywords: finite groups of uniformly bounded rank
zbMATH Keywords: numbers of generators
MaRDI profile type: Publication
full work available at URL: s000130050308
OpenAlex ID: W2077325469
DOI: 10.1007/S000130050308
description: scientific article; zbMATH DE number 1295478