Nonsolvable groups with an automorphism inverting many elements
type: Item
instance of: scholarly article
title: Nonsolvable groups with an automorphism inverting many elements
zbMATH Open document ID: 0645.20008
DOI: 10.1007/BF01190221
author: Walter M. Potter
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: Let G be a finite group, let \(\alpha\in Aut(G)\) and \(T(\alpha)=\{g\in G|\alpha (g)=g^{-1}\}\). \textit{H. Liebeck} and \textit{D. MacHale} [Math. Z. 124, 51-63 (1972; Zbl 0213.305)] have shown that if \(| T(\alpha)| >| G|\), then G is abelian. The present paper is concerned with the case \(| T(\alpha)| >| G| /4\). It is proved that the only simple nonabelian group satisfying this condition is \(A_ 5\). In this case \(\alpha\) can be choosen to be the inner automorphism induced by an involution and \(| T(\alpha)| =4| A_ 5| /15\). As a corollary, the following sufficient condition for solvability is obtained: G is solvable if it has an automorphism \(\alpha\) such that \(| T(\alpha)| >4| G| /15.\) The main result of the paper is the characterization of the nonsolvable groups G having an automorphism \(\alpha\) with \(| T(\alpha)| >| G| /4\). Such a group is isomorphic to the direct product \(A_ 5\times A\), where A is abelian, \(A\subset T(\alpha)\). Moreover, \(| T(\alpha)| =4| G| /15.\) It seems that the techniques employed in this paper could be used in corresponding problems concerning finite groups having an automorphism which squares of cubes relative large proportions of the elements of the group.
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 4051871
zbMATH Keywords: simple nonabelian group
zbMATH Keywords: \(A_ 5\)
zbMATH Keywords: solvability
zbMATH Keywords: nonsolvable
zbMATH Keywords: automorphism
reviewed by: Marian Deaconescu
MaRDI profile type: Publication
cites work: Q3708174
cites work: Endliche Gruppen I
description: scientific article; zbMATH DE number 4051871