Automorphisms with finite Reidemeister number in residually finite groups.
type: Item
instance of: scholarly article
title: Automorphisms with finite Reidemeister number in residually finite groups.
zbMATH Open document ID: 1161.20026
author: Enrico Jabara
published in: Journal of Algebra
publication date: 2008-12-10T00:00:00Z
review text: Let \(f\) be an automorphism of the group \(G\) of finite order. Say elements \(x\) and \(y\) of \(G\) are related if there exists \(g\) in \(G\) with \(gx=yg^f\). This defines an equivalence relation on \(G\). Assume also that its number of equivalence classes is finite. The author proves the following results. If \(f\) has prime order and if either \(G\) is residually finite or \(G\) is soluble and finitely generated, then \(G\) is nilpotent-by-finite. If \(G\) is linear and finitely generated, then \(G\) is soluble-by-finite. In the first result the prime restriction on \(f\) cannot be removed; the author gives finitely generated, metabelian counterexamples with \(f\) of order 4 and just 2 equivalence classes. The paper also contains a brief survey of where automorphisms (even endomorphisms) of this type have arisen previously in the literature.
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20H20
zbMATH DE Number: 5380145
zbMATH Keywords: automorphisms
zbMATH Keywords: residually finite groups
zbMATH Keywords: soluble groups
zbMATH Keywords: linear groups
reviewed by: B. A. F. Wehrfritz
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.09.001
OpenAlex ID: W2037940202
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DOI: 10.1016/J.JALGEBRA.2008.09.001
description: scientific article; zbMATH DE number 5380145