Solubility of finite groups admitting a certain fixed-point-free group of automorphisms
type: Item
instance of: scholarly article
title: Solubility of finite groups admitting a certain fixed-point-free group of automorphisms
zbMATH Open document ID: 0682.20013
DOI: 10.1016/0021-8693(89)90262-7
author: Yanyan Li
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: This paper is a further contribution to the conjecture that if the finite group G has a fixed-point free automorphism group A and \((| G|,| A|)=1\), then G is soluble. This was proved by Martineau when A is elementary Abelian and by Rickman when the order of A is the square of a prime. It is proved here when A is an Abelian r-group of type (2,1) and \(C_ G(\Omega_ 1(A))\) is of order a power of an odd prime. This is deduced from Theorem 1.1: if A is soluble, \((| G|,| A|)=1\) and the centre of A contains an elementary Abelian subgroup B of order \(r^ 2\) such that \(C_ G(b)\) is of order a power of an odd prime for every non-identity element b of B, then G is soluble. The above statement follows from this using the results of Martineau and Rickman. After a few preliminary remarks, the bulk of the paper is concerned with the properties of a minimal counterexample to Theorem 1.1.
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 4116793
zbMATH Keywords: solubility
zbMATH Keywords: fixed-point free automorphism group
zbMATH Keywords: minimal counterexample
reviewed by: Norman Blackburn
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90262-7
OpenAlex ID: W1988075926
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description: scientific article; zbMATH DE number 4116793