On groups satisfying a positive law in fixed points.
type: Item
instance of: scholarly article
title: On groups satisfying a positive law in fixed points.
zbMATH Open document ID: 1182.20032
author: Aline de Souza Lima
author: Pavel Shumyatsky
published in: Journal of Algebra
publication date: 2009-09-10T00:00:00Z
review text: The goal of the paper under review is to prove the following result. Let \(q\) be a prime. Suppose the elementary Abelian \(q\)-group \(A\) of order \(q^2\) acts on the finite \(q'\)-group \(G\) so that for any non-identity \(a,b\in A\) the subgroup \(\langle C_G(a),C_G(b)\rangle\) satisfies a law which is positive (i.e., it does not include inverses), of degree \(n\). Then \(G\) satisfies a positive law of degree bounded by a function of \(q\) and \(n\) only. This result is part of a large body of literature which studies the influence of the structure of the centralizer of a coprime action on the structure of a finite group. Celebrated results in this area include \textit{J.~G.~Thompson}'s proof that a finite group \(G\) admitting a fixed-point-free automorphism of prime order \(q\) is nilpotent [Proc. Natl. Acad. Sci. USA 45, 578-581 (1959; Zbl 0086.25101)], and G.~Higman's bound on the nilpotence class of such a \(G\) in terms of \(q\) alone.
reviewed by: Andrea Caranti
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20E36
zbMATH DE Number: 5602651
zbMATH Keywords: automorphisms of groups
zbMATH Keywords: associated Lie rings
zbMATH Keywords: positive laws in groups
zbMATH Keywords: coprime actions
zbMATH Keywords: centralizers of automorphisms
zbMATH Keywords: semigroup laws
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2009.01.010
OpenAlex ID: W1963480435
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DOI: 10.1016/J.JALGEBRA.2009.01.010
description: scientific article; zbMATH DE number 5602651