Hall \(\pi\)-subgroups which are direct products of non-Abelian simple groups
type: Item
instance of: scholarly article
title: Hall \(\pi\)-subgroups which are direct products of non-Abelian simple groups
zbMATH Open document ID: 0665.20010
DOI: 10.1016/0021-8693(89)90186-5
author: Pamela A. Ferguson
author: Patricia Kelley
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: The main result of this paper is Corollary B: If H is characteristically simple, G is simple, \(H<G\), and H is a Hall \(\pi\)-subgroup of G, then the inclusion is one of the following: (i) \(A_ 5<L_ 2(q)\), (ii) \(A_{p- 1}<A_ p\) for p prime, \(p\geq 7\), (iii) \(M_{22}<M_{23}\), (iv) \(PSp_ 6(2)<P\Omega_ 7(q)\), or (v) \(P\Omega^+_ 8(2)<P\Omega^+_ 8(q)\). In cases (i), (iv) and (v) there are of course restrictions on the possible values of q. Theorem A is a straightforward generalization to arbitrary finite groups G, but is too complicated to state here. Since every finite simple group has even order, it follows that \(| G:H|\) is odd. Now \textit{M. Liebeck} and \textit{J. Saxl} [J. Lond. Math. Soc., II. Ser. 31, 250-264 (1985; Zbl 0573.20004)] have completely classified the primitive permutation groups of odd degree. The proof of Corollary B now goes by considering the various cases in turn. In particular, it depends heavily on the classification of finite simple groups.
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 4087906
zbMATH Keywords: characteristically simple
zbMATH Keywords: Hall \(\pi\)-subgroup
zbMATH Keywords: primitive permutation groups
zbMATH Keywords: classification of finite simple groups
reviewed by: Robert A. Wilson
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90186-5
OpenAlex ID: W2036300243
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description: scientific article; zbMATH DE number 4087906