Toward a characterization of the sporadic groups of Suzuki and Thompson
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 analysis shows that \(p=3\) and gives a precise information on the structure of the centralizer of certain elements \(t\) where \(|t|=2\) or \(3\). Moreover one obtains a characterization of the sporadic groups \(Suz\) and \(Th\).
scientific article; zbMATH DE number 1868070