On groups of local characteristic \(p\). (Q2498868)

The structure of maximal 2-local subgroups of finite simple groups \(G\) is restricted. More precisely, if \(S\in\text{Syl}_2(G)\) and \(S\leq M\) is a maximal 2-local subgroup such that \(\Omega_1(Z(S))\ntrianglelefteq M\). The aim of the author is to prove a general result about the structure of maximal \(p\)-local subgroups in groups \(G\) of local characteristic \(p\). Let \(G\) be a finite group of local characteristic \(p\) and with a trivial \(p\)-radical. The author shows that for a maximal \(p\)-local subgroup \(H\) with \(S\leq H\), \(S\in\text{Syl}_p(G)\), one has three alternatives: (1) \(H\) is unique, (2) \(H\) has the property that the maximal normal elementary Abelian \(p\)-subgroup \(Y_H\leq H\) is a \(2F\)-module for \(H/C_H(Y_H)\) with a cubic offender \(A\), or (3) the dual module of \(Y_H\) is an \(F\)-module for \(H/C_H(Y_H)\). This result is motivated by some similar result for quasithin groups of even type [\textit{M. Aschbacher} and \textit{S. D. Smith}, The classification of quasithin groups. I. Math. Surv. Monogr. 111 AMS (2004; Zbl 1065.20023)].