Bounding the exponent of a finite group with automorphisms
type: Item
instance of: scholarly article
title: Bounding the exponent of a finite group with automorphisms
zbMATH Open document ID: 0923.20014
published in: Journal of Algebra
publication date: 1999-10-27T00:00:00Z
review text: The authors consider the following situation: (a) \(A\) is a noncyclic Abelian group of automorphisms of a finite group \(G\), \((|A|,|G|)=1\), (b) \(q\) is the smallest prime such that the Sylow-\(q\)-subgroup of \(A\) is noncyclic, (c) the least common multiple of the exponents of the centralizers \(C_G(a)\), with \(a\in A\) and non-identity, is \(n\). They show that the exponent of \(G\) is bounded in terms of \(q\) and \(n\) in this situation (Theorem 1). As a consequence they show that a torsion compact group possessing an elementary Abelian subgroup with finite centralizer has bounded exponent. The proof is via Lie algebra arguments.
reviewed by: Hermann Heineken
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 22C05
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 1271657
zbMATH Keywords: Abelian groups of automorphisms
zbMATH Keywords: finite groups
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: exponents
zbMATH Keywords: centralizers
zbMATH Keywords: torsion compact groups
zbMATH Keywords: Lie algebras
author: Evgenii I. Khukhro
author: Pavel Shumyatsky
MaRDI profile type: Publication
full work available at URL: jabr.1998.7609
OpenAlex ID: W2063016618
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DOI: 10.1006/JABR.1998.7609
description: scientific article; zbMATH DE number 1271657