Finite groups admitting a fixed-point-free automorphism group
type: Item
instance of: scholarly article
title: Finite groups admitting a fixed-point-free automorphism group
zbMATH Open document ID: 0835.20036
published in: Journal of Algebra
publication date: 1996-04-10T00:00:00Z
review text: Using the classification of finite simple groups, the author proves that if a finite group \(A\) acts fixed-point-freely on a finite group \(G\) and that either \(A\) is cyclic or \((|G|,|A|)=1\), then \(G\) is solvable. The proof is dependent on the following lemmas, each of interest in itself. (1) Let \(H\) be an \(A\)-invariant subgroup of \(G\). Then \(A\) acts fixed-point-freely on \(H\) and if \(H\trianglelefteq G\), fixed- point-freely on \(G/H\). (2) Suppose that \(A\) and \(G\) both act upon a finite set \(\Omega\). If \(G\) is transitive on \(\Omega\) and \((\omega g)^\alpha=\omega^\alpha g^\alpha\) for all \(\omega\in\Omega\), \(g\in G\), and \(\alpha \in A\), there is a unique element of \(\Omega\) fixed by \(A\). (3) If \(G\) has a cyclic Sylow \(p\)-subgroup, then \(G\) has a normal \(p\)-complement [observed first by \textit{F. Gross}: Can. J. Math. 20, 1300-1307 (1968; Zbl 0186.31803)].
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 783885
zbMATH Keywords: fixed-point-free automorphisms
zbMATH Keywords: finite groups
zbMATH Keywords: cyclic Sylow \(p\)- subgroups
zbMATH Keywords: normal \(p\)-complements
author: Peter J. Rowley
reviewed by: Homer F. Bechtell
MaRDI profile type: Publication
full work available at URL: jabr.1995.1148
OpenAlex ID: W2009103579
DOI: 10.1006/JABR.1995.1148
description: scientific article; zbMATH DE number 783885