Locally finite groups and their subgroups with small centralizers
type: Item
instance of: scholarly article
title: Locally finite groups and their subgroups with small centralizers
zbMATH Open document ID: 1362.20033
author: Kıvanç Ersoy
author: Mahmut Kuzucuoğlu
author: Pavel Shumyatsky
published in: Journal of Algebra
publication date: 2017-05-12T00:00:00Z
review text: A group \(G\) is almost locally soluble if \(G\) contains a locally soluble subgroup of finite index. \textit{B. Hartley} [J. Lond. Math. Soc., II. Ser. 37, No. 3, 421--436 (1988; Zbl 0619.20018)] proved that if a locally finite group contains an element of prime-power order with Chernikov centralizer, then it is almost locally soluble. The main result in this paper is Theorem 1.1: Let \(p\) be a prime and \(G\) a locally finite group containing an elementary abelian \(p\)-subgroup \(A\) of rank at least 3 such that \(C_{G}(A)\) is Chernikov and \(C_{G}(a)\) involves no infinite simple groups for any \(a \in A^{\#}\). Then, \(G\) is almost locally soluble. In view of the aforementioned result of Hartley [loc. cit.], the theorem remains valid also in the case where \(A\) is of prime order. On the other hand, Theorem 1.1 is no longer valid when \(A\) is of rank two. More precisely, the authors are able to prove Theorem 1.2: An infinite simple locally finite group \(G\) admits an elementary abelian \(p\)-group of automorphisms \(A\) such that \(C_{G}(A)\) is Chernikov and \(C_{G}(a)\) involves no infinite simple groups for any \(a\in A^{\#}\) if and only if \(G\) is isomorphic to \(\mathrm{PSL}_{p}(k)\) for some locally finite field \(k\) of characteristic different from \(p\) and \(A\) has order \(p^{2}\).
reviewed by: Enrico Jabara
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 6714591
zbMATH Keywords: locally finite groups
zbMATH Keywords: Chernikov centralizer
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2017.02.018
OpenAlex ID: W2592079542
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DOI: 10.1016/J.JALGEBRA.2017.02.018
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description: scientific article; zbMATH DE number 6714591