Finite groups with an automorphism of prime order whose centralizer has small rank.
type: Item
instance of: scholarly article
title: Finite groups with an automorphism of prime order whose centralizer has small rank.
zbMATH Open document ID: 1108.20019
published in: Journal of Algebra
publication date: 2006-12-07T00:00:00Z
review text: Let \(\varphi\) be an automorphism of prime order \(p\) of a finite group \(G\), and \(C_G(\varphi)\) its fixed point subgroup. By the classical theorem of \textit{J. Thompson}, [Proc. Natl. Acad. Sci. USA 45, 578-581 (1959; Zbl 0086.25101)], if \(C_G(\varphi)=1\) (that is, \(\varphi\) is regular), then \(G\) is nilpotent. It is natural to expect that if \(\varphi\) is ``almost regular'', then \(G\) must be ``almost nilpotent''. The ``almost regular'' result of \textit{P. Fong}, [Osaka J. Math. 13, 483-489 (1976; Zbl 0372.20010)], and Hartley-Meixner-Pettet, [see \textit{B. Hartley, T. Meixner}, Arch. Math. 36, 211-213 (1981; Zbl 0447.20014) and \textit{M. R. Pettet}, J. Algebra 72, 404-412 (1981; Zbl 0473.20018)], gives the existence of a nilpotent subgroup of index bounded in terms of \(p\) and \(|C_G(\varphi)|\). In this paper, the authors prove an analogue of the Hartley-Meixner-Pettet theorem for soluble groups and an analogue of Fong's theorem in the case where the orders of \(G\) and \(\varphi\) are coprime (the examples in this paper show that there can be no rank analogue of Fong's theorem if the orders of \(G\) and \(\varphi\) are not coprime). Three rather complicated theorems are proved in the present paper.
reviewed by: Shi Wujie
Mathematics Subject Classification ID: 20D45
zbMATH DE Number: 5078117
zbMATH Keywords: regular automorphisms
zbMATH Keywords: ranks
zbMATH Keywords: Hall-Higman-type theorems
zbMATH Keywords: powerful \(p\)-groups
zbMATH Keywords: almost nilpotent groups
author: Evgenii I. Khukhro
author: V. D. Mazurov
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.02.039
OpenAlex ID: W2062818653
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DOI: 10.1016/J.JALGEBRA.2006.02.039
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description: scientific article; zbMATH DE number 5078117