Automorphisms of finite groups of bounded rank
type: Item
instance of: scholarly article
title: Automorphisms of finite groups of bounded rank
zbMATH Open document ID: 0817.20026
DOI: 10.1007/BF02808121
author: Aner Shalev
published in: Israel Journal of Mathematics
publication date: 1995-07-31T00:00:00Z
review text: The author proves that, if \(G\) is a finite group of rank \(r\) admitting an automorphism \(\alpha\) with \(m\) fixed points, then \(G\) has a soluble characteristic subgroup \(H\) whose index is bounded by a function of \(m\) and \(r\). Moreover, if the orders of \(G\) and \(\alpha\) are coprime, then the derived length of \(H\) is also bounded by a function of \(m\) and \(r\). In order to show that the derived length of \(H\) is bounded, the author reduces to a Lie-theoretic problem, and this reduction is based on the construction of a particular Lie ring from a certain type of \(p\)-group. Concerning Lie rings, he proves that if \(L\) is a Lie ring of finite rank \(r\) admitting a semisimple fixed point-free automorphism \(\alpha\), then \(L\) is soluble with derived length at most \(2^{r-1} - 1\).
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 17B60
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F40
zbMATH DE Number: 446719
zbMATH Keywords: finite groups
zbMATH Keywords: soluble characteristic subgroups
zbMATH Keywords: index
zbMATH Keywords: orders
zbMATH Keywords: derived length
zbMATH Keywords: Lie rings
zbMATH Keywords: fixed point-free automorphisms
reviewed by: Silvana Franciosi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 446719