A general Brauer-Fowler theorem and centralizers in locally finite groups
type: Item
instance of: scholarly article
title: A general Brauer-Fowler theorem and centralizers in locally finite groups
zbMATH Open document ID: 0713.20023
author: Brian Hartley
published in: Pacific Journal of Mathematics
publication date: 1992-01-01T00:00:00Z
review text: A classical theorem of Brauer and Fowler, proved by quite elementary methods, states that the order of a finite non-abelian simple group G is bounded in terms of the order of the centralizer of any involution in G. Here we use the classification of finite simple groups to verify that the order of such a G is bounded in terms of the order of any automorphism \(\alpha\) of G and the number of fixed points of \(\alpha\). It follows easily that if a locally finite group G contains an element x with finite centralizer, then it has a locally soluble subgroup H of finite index. In particular, if G itself is simple, then it is finite. If, further, x has prime power order \(p^ n\), then we may take \(H=O_{p'p}(H)=F_ n(H)\), where \((F_ i(H))\) is the Hirsch-Plotkin series of H. As a further application of the techniques used to prove the above, we improve a result of the author [J. Lond. Math. Soc., II. Ser. 37, 421-436 (1988; Zbl 0619.20018)]. We show that if G is an infinite simple group of Lie type over a locally finite field K of characteristic p, and \(\alpha\) is an automorphism of G of finite order not divisible by p, then there exist infinitely many primes q such that \(\alpha\) fixes an element of order q of G. This was proved previously under restrictions on the order of \(\alpha\).
reviewed by: Brian Hartley
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 4173236
zbMATH Keywords: order of a finite non-abelian simple group
zbMATH Keywords: order of the centralizer
zbMATH Keywords: involution
zbMATH Keywords: classification of finite simple groups
zbMATH Keywords: automorphism
zbMATH Keywords: number of fixed points
zbMATH Keywords: locally finite group
zbMATH Keywords: finite centralizer
zbMATH Keywords: locally soluble subgroup H of finite index
zbMATH Keywords: Hirsch-Plotkin series
zbMATH Keywords: infinite simple group of Lie type
zbMATH Keywords: locally finite field
MaRDI profile type: Publication
full work available at URL: pjm.1992.152.101
OpenAlex ID: W2075031783
DOI: 10.2140/PJM.1992.152.101
description: scientific article; zbMATH DE number 4173236