On almost regular automorphisms of finite \(p\)-groups
type: Item
instance of: scholarly article
title: On almost regular automorphisms of finite \(p\)-groups
zbMATH Open document ID: 0968.20011
published in: Advances in Mathematics
publication date: 2001-02-13T00:00:00Z
review text: This remarkable paper contains a positive solution of the last major open problem in the theory of almost regular \(p\)-automorphisms of finite \(p\)-groups (based on the corresponding result on Lie rings). Suppose that a finite \(p\)-group \(P\) admits an automorphism \(\varphi\) of order \(p^k\) having exactly \(p^m\) fixed points. What restrictions on the structure of \(P\) can be obtained in terms of \(p^k\) and \(p^m\)? Though a \(p\)-automorphism of a finite \(p\)-group cannot be regular, i.~e. without non-trivial fixed points, it is theorems of G.~Higman and V.~A.~Kreknin on regular automorphisms of Lie rings that are instrumental in this kind of problems. In the case of \(|\varphi|=p\), \textit{J.~L.~Alperin} [Proc. Am. Math. Soc. 13, 175-180 (1962; Zbl 0104.02801)] used Higman's theorem and the associated Lie rings to show that the derived length of \(P\) is bounded in terms of \(p\) and \(m\) (for short, ``\((p,m)\)-bounded''). \textit{E.~I.~Khukhro} [Mat. Zametki 38, No. 5, 652-657 (1985; Zbl 0598.20024)] proved that, moreover, \(P\) then has a subgroup of \((p,m)\)-bounded index which is nilpotent of \(p\)-bounded class. In the general case of \(|\varphi|=p^k\), \textit{A.~Shalev} [J. Algebra 157, No. 1, 271-282 (1993; Zbl 0797.20023)] used Kreknin's theorem and the ``\(p\)-adic'' construction of a Lie ring for uniformly powerful \(p\)-groups to show that the derived length of \(P\) is \((p,k,m)\)-bounded. Using the Mal'cev correspondence as well as powerful \(p\)-groups, \textit{E.~I.~Khukhro} [Mat. Sb. 184, No. 12, 53-64 (1993; Zbl 0836.20018)] proved that, moreover, \(P\) has a subgroup of \((p,k,m)\)-bounded index which is soluble of \(p^k\)-bounded derived length. Results on almost regular \(p\)-automorphisms of finite \(p\)-groups are related to the theory of \(p\)-groups of maximal class and of given coclass (C.~R.~Leedham-Green, S.~McKay, A.~Mann, M.~Newman, S.~Donkin, A.~Shalev, E.~I.~Zelmanov and others). The theory of \(p\)-groups of maximal c…
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 20D45
zbMATH DE Number: 1507792
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: \(p\)-automorphisms
zbMATH Keywords: fixed points
zbMATH Keywords: \(p\)-adic integers
zbMATH Keywords: Lie rings
zbMATH Keywords: nilpotency classes
zbMATH Keywords: derived lengths
zbMATH Keywords: almost regular automorphisms
zbMATH Keywords: powerful \(p\)-groups
author: Andrei Jaikin-Zapirain
reviewed by: Evgenii I. Khukhro
MaRDI profile type: Publication
full work available at URL: aima.1999.1911
OpenAlex ID: W2082688995
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DOI: 10.1006/AIMA.1999.1911
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description: scientific article; zbMATH DE number 1507792