{
  "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 $\\mathrm{PSU}_n(q)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.19105",
  "author_total": 1,
  "authors": [
    {
      "name": "Patricia Medina Capilla"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5ppncsdctae64.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.19105",
  "fulltext_url": null,
  "id": "5ppncsdctae64",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.19105",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7203892929",
  "pdf_url": "https://arxiv.org/pdf/2608.19105",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5ppncsdctae64.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2608.19105",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.19105"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The structure and generation of the second maximal subgroups of the almost simple groups with alternating, classical or sporadic socle",
  "title_text": "The structure and generation of the second maximal subgroups of the almost simple groups with alternating, classical or sporadic socle",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/5ppncsdctae64",
  "venue": "arXiv",
  "versions": []
}