Groups of even type which are not of even characteristic. I
At present, there are two strategies for revising the classification of the finite simple groups. \textit{D. Gorenstein} et al. work with groups of even type (see Definition 21.3 from their book [The classification of the finite simple groups. Providence, RI: AMS, American Mathematical Society (1994; Zbl 0816.20016)]), this reflects that semisimple elements are the primary focus of the investigation. \textit{U. Meierfrankenfeld} et al. (see, for example, their book [``The structure theorem for finite groups with a large \(p\)-subgroup'', Mem. Am. Math. Soc. 1147, 356 p. (2016; \url{doi:10.1090/memo/1147})]) follow a different strategy. They work with groups of even characteristic and use unipotent methods, notably the amalgam method. A finite group \(G\) is said to be of even characteristic, if for a Sylow 2-subgroup \(S\) and all nontrivial 2-local subgroups \(H\) of \(G\) with \(S\leq H\), we have that \(C_G(O_2(H))\leq O_2(H)\). In the given paper, the authors clarify the relationship between the two definitions and thus make the amalgam method available for use in the classification of groups of even type. The main theorem of both parts of this paper is: If \(G\) is a finite simple of even type and any simple factor of any nontrivial 2-local subgroup of \(G\) is known, then either \(G\) is of even characteristic or \(G\) is isomorphic to \(J_1\), \(\mathrm{Co}_3\), \(A_{12}\), \(P\Omega_7(3)\) or \(P\Omega_8^-(3)\). This theorem is a generalization of Chapter 16 of the work by \textit{M. Aschbacher} and \textit{S. D. Smith} [The classification of quasithin groups. I: Structure of strongly quasithin \(\mathcal K\)-groups. Providence, RI: American Mathematical Society (AMS) (2004; Zbl 1065.20023); The classification of quasithin groups. II: Main theorems: the classification of simple QTKE-groups. Providence, RI: American Mathematical Society (AMS) (2004; Zbl 1065.20024)]. In this first part of the paper, it is shown that if \(G\) is a counterexample to the main theorem then there is a 2-central involution \(z\) in \(G\) such that \(C_G(z)\) possesses a standard subgroup \(L\) and \(C_G(L)\) is cyclic.
scientific article; zbMATH DE number 6627901
scientific article