On groups that are residually of finite rank
type: Item
instance of: scholarly article
title: On groups that are residually of finite rank
zbMATH Open document ID: 0914.20027
DOI: 10.1007/BF02764002
author: Martyn Dixon
author: Martin J. Evans
author: Howard Smith
published in: Israel Journal of Mathematics
publication date: 1999-06-17T00:00:00Z
review text: A group has rank \(r\), \(r\) a positive integer, if every finitely generated subgroup of \(G\) can be generated by \(r\) elements, and if no smaller integer has this property. The authors study groups that are residually of rank \(r\). For example, Theorem 1 states that a periodic group that is residually (locally finite and of rank \(r\)), is locally finite. Theorem 2 is a (partial) structure result for groups that are residually (locally soluble and of rank \(r\)). Theorem 4 implies that a locally soluble group is hyperabelian whenever it is residually of rank \(r\). The authors illustrate their results with examples. For example, the standard wreath product of two infinite cyclic groups has infinite rank, but is residually of rank 2 (weakish form of Theorem 8).
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E25
zbMATH DE Number: 1239657
zbMATH Keywords: residually of finite rank groups
zbMATH Keywords: finite rank groups
zbMATH Keywords: periodic groups
zbMATH Keywords: locally soluble groups
zbMATH Keywords: finitely generated groups
reviewed by: B. A. F. Wehrfritz
MaRDI profile type: Publication
cites work: Q3123478
cites work: Theorem on groups of finite special rank
cites work: Q3997907
cites work: Locally (soluble-by-finite) groups of finite rank
cites work: On Burnside's problem on periodic groups
cites work: Residual Properties of Infinite Soluble Groups
cites work: Q5730935
cites work: On Finitely Generated Soluble Groups With No Large Wreath Product Sections
cites work: Some remarks on Sylow subgroups of general linear groups
cites work: Q5654235
cites work: A footnote on residually finite groups
cites work: Q5639880
cites work: Q5676405
cites work: Two-Generator conditions for Residually Finite Groups
full work available at URL: https://doi.org/10.1007/bf02764002
OpenAlex ID: W2041306823
description: scientific article; zbMATH DE number 1239657