Positive laws in fixed points of automorphisms of finite groups.
type: Item
instance of: scholarly article
title: Positive laws in fixed points of automorphisms of finite groups.
zbMATH Open document ID: 1235.20024
published in: Journal of Pure and Applied Algebra
publication date: 2011-10-21T00:00:00Z
review text: Suppose that a finite group \(G\) admits a Frobenius group of automorphisms \(FH\) with kernel \(F\) and complement \(H\) such that the kernel \(F\) is cyclic and acts fixed-point-freely, that is, \(C_G(F)=1\). It is proved that if \(C_G(H)\) satisfies a positive law of degree \(k\), then \(G\) also satisfies a positive law of degree bounded in terms of \(k\) and \(|FH|\). Here, a positive law is a group law of the form \(u\equiv v\), where \(u\) and \(v\) are group words involving only positive powers of the variables. The proofs in the present paper, many of them by Lie ring methods, combine in quite a non-trivial way some ideas and techniques from the recent paper by \textit{E. Khukhro, N. Yu. Makarenko} and \textit{P. Shumyatsky}, [Frobenius groups of automorphisms and their fixed points, \url{arXiv:1010.0343v1}], where it was proved that the exponent of \(G\) is bounded in terms of the exponent of \(C_G(H)\) and \(|FH|\), and if \(C_G(H)\) is nilpotent, then \(G\) is also nilpotent of nilpotency class bounded in terms of the nilpotency class of \(C_G(H)\) and \(|H|\). Note that a finite group satisfying a positive law of degree \(k\) is an extension of a nilpotent group of \(k\)-bounded class by a group of \(k\)-bounded exponent [\textit{R. G. Burns, O. Macedońska, Yu. Medvedev}, J. Algebra 195, No. 2, 510-525 (1997; Zbl 0886.20022)]; the converse is also true: an extension of a nilpotent group of class \(c\) by a group of exponent \(e\) satisfies a positive law of degree bounded in terms of \(c\) and \(e\) [\textit{A. I. Mal'tsev}, Ivanov. Gos. Ped. Inst., Uchenye Zap., Fiz.-Mat. Nauki 4, 107-111 (1953; Zbl 0087.25501)]. In a broader context, for a finite group \(G\) admitting a Frobenius group of automorphisms \(FH\) with fixed-point-free kernel \(F\), a few other results were recently obtained on the structure of \(G\) in terms of \(C_G(H)\) and \(H\) [e.g., \textit{N. Yu. Makarenko, P. Shumyatsky}, Proc. Am. Math. Soc. 138, No. 10, 3425-3436 (2010; Zbl 1214…
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 5960886
zbMATH Keywords: positive laws
zbMATH Keywords: Frobenius groups of automorphisms
zbMATH Keywords: soluble finite groups
zbMATH Keywords: Lie rings
zbMATH Keywords: fixed-point-free automorphisms
zbMATH Keywords: nilpotency classes
zbMATH Keywords: exponents
author: Pavel Shumyatsky
reviewed by: Evgenii I. Khukhro
MaRDI profile type: Publication
full work available at URL: j.jpaa.2011.02.018
OpenAlex ID: W2051238533
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DOI: 10.1016/J.JPAA.2011.02.018
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description: scientific article; zbMATH DE number 5960886