On locally finite groups with a small centralizer of a four-subgroup.
type: Item
instance of: scholarly article
title: On locally finite groups with a small centralizer of a four-subgroup.
zbMATH Open document ID: 1234.20046
author: Emanuela Romano
author: Pavel Shumyatsky
published in: Archiv der Mathematik
publication date: 2011-08-30T00:00:00Z
review text: In this interesting paper, the authors prove the following, see 1.1, 1.2 and 4.4. Let \(G\) be a locally finite group containing a Klein-4 subgroup \(V\) such that its centralizer \(C_G(V)\) in \(G\) is finite. If \(V\) contains an element \(v\) whose centralizer in \(G\) has finite exponent, then \(G\) is (locally soluble)-by-finite and \([G,V]\) has finite index in \(G\). If \(V\) contains distinct involutions \(v\) and \(w\) such that \(C_G(v)\) and \(C_G(w)\) both have finite exponent, then \([G,V]'\) has finite exponent. Thus in the latter case we have a quite tight possible structure for \(G\).
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20F28
zbMATH DE Number: 5944170
zbMATH Keywords: locally finite groups
zbMATH Keywords: finite centralizers
zbMATH Keywords: 4-groups of automorphisms
zbMATH Keywords: groups of finite exponent
reviewed by: B. A. F. Wehrfritz
MaRDI profile type: Publication
full work available at URL: s00013-011-0273-y
OpenAlex ID: W1969645640
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DOI: 10.1007/S00013-011-0273-Y
description: scientific article; zbMATH DE number 5944170