具有交错、古典或零散基座的几乎单群的第二极大子群的结构与生成
The structure and generation of the second maximal subgroups of the almost simple groups with alternating, classical or sporadic socle
- University of Warwick（华威大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对基座为交错、古典或零散群的几乎单群，修正了其非抛物极大子群生成元数量的分类，将生成元界优化至最优值，确定了相关主因子并改进了已有结论。
AI中文摘要：
设 G 为基座是交错群、古典群或零散群的几乎单群，H 为 G 的非抛物极大子群。我们证明，H 的任意极大子群 M 最多可由 7 个元素生成，且当 G 的基座为交错群或古典群时，该界是紧的；这改进了 Burness、Liebeck 和 Shalev 给出的 12 的界。当基座为零散群时，最多需要 5 个生成元，且该界同样是最优的。证明依赖于对 H 和 M 的详细结构分析，尤其当 H 是 wreath 积的子群时。特别地，我们确定了 M 的主因子，随后利用 crown 理论对其生成元数量进行了界定。我们还修正了 Lucchini、Marion 和 Tracey 建立的关于需要超过 3 个生成元的几乎单群极大子群 H 的分类。他们给出的 5 个生成元的界仍然有效，但他们的分类遗漏了若干对 (G,H)，包括 G 的基座为 PSUₙ(q) 的情形。
英文摘要：
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 $\mathrm{PSU}_n(q)$.
