On orbits of automorphism groups. II.
type: Item
instance of: scholarly article
title: On orbits of automorphism groups. II.
zbMATH Open document ID: 1179.20020
author: Marian Deaconescu
author: G. L. Walls
published in: Archiv der Mathematik
publication date: 2009-06-16T00:00:00Z
review text: Let \(A\) be a group of automorphisms of a finite group \(G\), let \(C_G(A)\) be the fixed point subgroup, and \(k_A(G)\) the number of \(A\)-orbits in \(G\). It is proved that if \(|C_G(A)|\) is coprime to \(k_A(G)\), then \(C_G(A)\subseteq [G,A]\) and the core of \(C_G(A)\) in \(G\) is contained in the centre of \([G,A]\). An automorphism \(\alpha\) is said to be class-avoiding, if \(x^\alpha\) belongs to the conjugacy class of \(x\) in \(G\) only if \(x^\alpha=x\). Any fixed-point free automorphism is class-avoiding by a result of Glauberman. Extending (and using) Thompson's celebrated theorem, the authors prove that if a class avoiding automorphism \(\alpha\in\Aut\,G\) of prime order coprime to \(|G|\) is such that \(C_G(\alpha)\leq[G,\alpha]\), then \(G\) is nilpotent. Similarly, based on the classification, solubility of \(G\) is derived for any class-avoiding automorphism \(\alpha\) if \((|\alpha|,|G|)=1\) and \(C_G(\alpha)\leq[G,\alpha]\). For part I cf. Sib. Mat. Zh. 46, No. 3, 533-537 (2005); translation in Sib. Math. J. 46, No. 3, 413-416 (2005; Zbl 1096.20025).
reviewed by: Evgenii I. Khukhro
Mathematics Subject Classification ID: 20D45
zbMATH DE Number: 5565247
zbMATH Keywords: automorphisms
zbMATH Keywords: finite groups
zbMATH Keywords: nilpotence
zbMATH Keywords: fixed points
zbMATH Keywords: group actions
zbMATH Keywords: stabilizers
zbMATH Keywords: numbers of orbits
zbMATH Keywords: automorphism groups
zbMATH Keywords: fixed point subgroups
zbMATH Keywords: class-avoiding automorphisms
zbMATH Keywords: solubility
MaRDI profile type: Publication
full work available at URL: s00013-009-3082-9
OpenAlex ID: W2006185950
DOI: 10.1007/S00013-009-3082-9
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description: scientific article; zbMATH DE number 5565247