Residually finite subgroups of some countable McLain groups.
type: Item
instance of: scholarly article
title: Residually finite subgroups of some countable McLain groups.
zbMATH Open document ID: 1195.20036
author: Giovanni Cutolo
author: Howard Smith
published in: Archiv der Mathematik
publication date: 2010-01-08T00:00:00Z
review text: The authors prove that certain torsion-free McLain groups have subgroups of periodic index that are residually finite. It was proven [\textit{M. R. Dixon, M. J. Evans} and \textit{H. Smith}, J. Algebra 320, No. 1, 81-85 (2008; Zbl 1160.20023)] that a countable torsion-free locally nilpotent group \(G\) that is also soluble has a residually finite subgroup \(J\) of periodic index in \(G\), that is, whose isolator in \(G\) is \(G\) itself. It also has been proven [in \textit{M. R. Dixon, M. J. Evans} and \textit{H. Smith}, J. Group Theory 9, No. 3, 383-395 (2006; Zbl 1120.20030)] that any countable group \(G\) that embeds in a simple group that is locally (soluble-by-finite) must at least have a residually finite subgroup of periodic index. The authors of the article under review describe the following important property of McLain groups. Theorem. Let \(A\) be a commutative ring with identity whose additive group has finite torsion-free rank and let \(R\) be a ring embeddable in a polynomial ring on a (possibly infinite) set of commuting indeterminates over \(A\). Then the McLain group \(M(\mathsf Q,R)\) has a residually finite subgroup of periodic index.
reviewed by: Igor Subbotin
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20H25
zbMATH DE Number: 5655339
zbMATH Keywords: McLain groups
zbMATH Keywords: residually finite groups
zbMATH Keywords: locally nilpotent groups
zbMATH Keywords: isolators
zbMATH Keywords: subgroups of periodic index
MaRDI profile type: Publication
full work available at URL: s00013-009-0061-0
OpenAlex ID: W1971341365
cites work: Embedding groups in locally (soluble-by-finite) simple groups
cites work: Residually finite subgroups of locally nilpotent groups.
cites work: Q5673086
cites work: Q5612538
cites work: Q5654235
DOI: 10.1007/S00013-009-0061-0
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description: scientific article; zbMATH DE number 5655339