Almost regular automorphisms
type: Item
instance of: scholarly article
title: Almost regular automorphisms
zbMATH Open document ID: 0956.20010
DOI: 10.1007/BF02366346
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-08-16T00:00:00Z
review text: This survey gives a detailed description of the theory of finite nilpotent groups admitting an automorphism with exactly \(m\) fixed points. The paper is divided into three sections. In section 1 the author discusses applications of theorems on Lie rings with regular automorphisms to the theory of nilpotent groups with regular automorphisms. Besides of the theorems of G. Higman, V. A. Kreknin and A. I. Kostrikin this section contains an account of Lie-ring methods such as the associated Lie ring and the Mal'cev correspondence. Section 2 is devoted to the coprime case where the order of the automorphism is coprime to the order of the group. The theorem of E. I. Khukhro on nilpotent groups with automorphism of prime order is cited as well as some of its applications and extensions. Section 3 deals with the situation where a nilpotent \(p\)-group admits an automorphism of order \(p^n\) with exactly \(p^m\) fixed points. The Alperin-Khukhro theorem, Khukhro-Shalev theorem, Medvedev's theorem and connections with \(p\)-groups of given coclass are discussed. Many results are provided with proofs or the basic ideas of the proofs. The reader can find open problems in this field.
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20-02
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 17-02
Mathematics Subject Classification ID: 17B40
Mathematics Subject Classification ID: 17B60
zbMATH DE Number: 1315225
zbMATH Keywords: almost regular automorphisms
zbMATH Keywords: finite nilpotent groups
zbMATH Keywords: automorphisms of prime order
zbMATH Keywords: Lie rings with regular automorphisms
zbMATH Keywords: finite \(p\)-groups
author: Evgenii I. Khukhro
reviewed by: N. Yu. Makarenko
MaRDI profile type: Publication
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