Toward a characterization of the sporadic groups of Suzuki and Thompson
type: Item
instance of: scholarly article
title: Toward a characterization of the sporadic groups of Suzuki and Thompson
zbMATH Open document ID: 1028.20011
DOI: 10.1016/S0021-8693(02)00529-X
author: Richard Lyons
author: Inna Capdeboscq
author: Ronald Solomon
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: A finite group \(G\) is of characteristic \(p\)-type (\(p\) a prime) if the \(p\)-rank \(m_p(G)\) of \(G\) is at least \(2\) and \(F^*(H)=O_p(H)\) for every \(p\)-local subgroup \(H\) of \(G\). The Klinger-Mason theorem [\textit{K. Klinger} and \textit{G. Mason}, J. Algebra 37, 362-375 (1975; Zbl 0325.20011)] states that a group which is of characteristic 2-type and of characteristic \(p\)-type (\(p>2\)) has the property that the \(p\)-rank of each \(2\)-local subgroup is at most \(2\). The analysis of such groups is therefore directed to the study of quasithin groups. The classification of quasithin groups was obtained recently with a second generation proof by \textit{M. Aschbacher} and \textit{S. D. Smith} [http://www.math.uic.edu/\(^\sim\)smiths/papers/quasithin]. In the paper under review the authors weaken the hypothesis of characteristic \(p\)-type in order to obtain a generalization of the Klinger-Mason theorem which among other groups characterizes the sporadic simple Suzuki group \(Suz\) and the sporadic simple Thompson group \(Th\). The intention of the authors is to apply this result in the revision project of \textit{D. Gorenstein}, \textit{R. Lyons} and \textit{R. Solomon} [``The classification of finite simple groups'', AMS monographs, Vol. 40, No. 1-5 (1994; Zbl 0816.20016, 1996; Zbl 0836.20011, 1998; Zbl 0890.20012, 1999; Zbl 0922.20021, 2002; Zbl 1006.20012)]. In the introduction to the paper under review one finds a precise explanation of the nature of this application. The authors call a group \(G\) of \(\widetilde p\)-type if \(m_p(G)\geq 3\), \(O_{p'}(H)=1\) for any \(p\)-local subgroup \(H\) of \(G\) and if \(K\) is a component of such a local subgroup \(H\) then \(K\in\text{Chev}(p)\) but \(K\neq 3A_6\). The authors investigate finite, \(\mathcal K\)-proper groups \(G\) of \(\widetilde 2\)-type and \(\widetilde p\)-type, \(p\) odd. Here \(\mathcal K\)-proper means that all composition factors of proper subgroups are known simple groups. This …
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20D08
zbMATH DE Number: 1868070
zbMATH Keywords: \(p\)-local subgroups
zbMATH Keywords: Klinger-Mason theorem
zbMATH Keywords: quasithin groups
zbMATH Keywords: characteristic \(p\)-type groups
zbMATH Keywords: sporadic simple Suzuki group
zbMATH Keywords: sporadic simple Thompson group
zbMATH Keywords: composition factors
zbMATH Keywords: simple groups
zbMATH Keywords: centralizers
reviewed by: Ulrich Dempwolff
MaRDI profile type: Publication
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full work available at URL: s0021-8693(02)00529-x
OpenAlex ID: W2071865629
description: scientific article; zbMATH DE number 1868070