The structure and generation of the second maximal subgroups of the almost simple groups with alternating, classical or sporadic socle

Abstract

Let $G$ be an almost simple group whose socle is an alternating, classical, or sporadic group, and let $H$ be a non-parabolic maximal subgroup of $G$. We prove that any maximal subgroup $M$ of $H$ can be generated by at most $7$ elements, and that this bound is sharp when the socle of $G$ is alternating or classical; this improves the bound of $12$ due to Burness, Liebeck and Shalev. When the socle is sporadic, at most $5$ generators suffice, and this is again best possible. The proof relies upon a detailed structural analysis of $H$ and $M$, especially when $H$ is a subgroup of a wreath product. In particular, we determine the chief factors of $M$ and subsequently bound its number of generators using the theory of crowns. We also correct the classification of maximal subgroups $H$ of almost simple groups requiring more than three generators, established by Lucchini, Marion and Tracey. Their bound of five generators remains valid, but their classification is missing several pairs $(G,H)$, including cases in which $G$ has socle $\mathrm{PSU}_n(q)$.