{
  "abstract": "We prove that every finite group that is the product of every pair of its non-conjugate maximal subgroups is soluble, answering Problem 10.34 of the Kourovka Notebook. The almost-simple case was proved by Tikhonenko and Tyutyanov. We treat the remaining minimal-counterexample branch, where the unique minimal normal subgroup has the form S^k, with S nonabelian simple and k >= 2. Two automorphism-stable coordinate subgroup classes satisfying a maximal-supplement criterion produce non-conjugate maximal supplements. If the factorization hypothesis held, their product would impose a p-adic divisibility requirement growing linearly with k, while the quotient contributes only coordinate outer automorphisms and a factor dividing k!. A fixed valuation gap therefore excludes every k >= 2 at once. Suitable subgroup classes are constructed uniformly across the infinite families of finite simple groups using parabolic, torus, and primitive-prime-divisor arguments; stable flag parabolics handle graph fusion, while GAP certificates cover designated finite and sporadic cases. The resulting all-k obstruction provides a reusable mechanism for eliminating direct-power socles in finite-group factorization problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.02970",
  "author_total": 1,
  "authors": [
    {
      "name": "Richie Sater"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/82e15f3rektew.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.02970",
  "fulltext_url": null,
  "id": "82e15f3rektew",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.02970",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7172490358",
  "pdf_url": "https://arxiv.org/pdf/2608.02970",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/82e15f3rektew.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.02970",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.02970"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Finite groups that are the product of every pair of non-conjugate maximal subgroups are soluble",
  "title_text": "Finite groups that are the product of every pair of non-conjugate maximal subgroups are soluble",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/82e15f3rektew",
  "venue": "arXiv",
  "versions": []
}