{
  "schema_version": 1,
  "problem_number": "KOU-19.82",
  "title": "On Shumyatsky's Question on Soluble Subgroups and the Generalized Fitting Height: A Partial Answer for Groups with Chief Factors of Type PSL(2,q) and Sz(q)",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let h*(G) be the generalized Fitting height of a finite group G, and h(K) the Fitting height of a soluble group K. P. Shumyatsky asked (Kourovka Notebook, Problem 19.82) whether every finite group G has a soluble subgroup K with h*(G) = h(K), that is, whether h*(G) never exceeds the largest Fitting height s(G) of a soluble subgroup; Khukhro and Shumyatsky proved λ(G) ≤ s(G) for the nonsoluble length λ(G), which is at most h*(G) (Israel J. Math. 2015). We give a partial answer: yes for every finite group whose nonabelian chief factors are of type PSL(2,q) (q ≥ 4) or Sz(q), provided that no PSL(2,q)-factor with q odd and q ≡ ±1 (mod 8) induces a group containing PGL(2,q); the general question stays open. For these groups, if O_p(G) = 1, some soluble K has h(K/O_p(K)) ≥ h*(G). Extending the method of Khukhro and Shumyatsky, we exclude a minimal counterexample by a witness system, a condition on the automorphisms induced on a minimal normal subgroup S^m, and construct witness systems for PSL(2,q) and Sz(q) from unipotent subgroups, Zsigmondy primes and the Lang–Steinberg theorem. The excluded case is a limit of the method, not a counterexample; the criterion is only sufficient and fails for some simple groups; A_n (n ≥ 7), the other groups of Lie type and the sporadic groups are not treated. No printed answer to the question was found. This is an unrefereed note.",
  "result_type": "PARTIAL_RESULT",
  "categories": [
    "math.GR"
  ],
  "keywords": [
    "generalized Fitting height",
    "Fitting height",
    "soluble subgroups",
    "quasinilpotent groups",
    "chief factors",
    "PSL(2,q)",
    "Suzuki groups",
    "Zsigmondy primes",
    "Lang-Steinberg theorem",
    "partial answer",
    "Kourovka Notebook problems",
    "UnsolvedMath",
    "KOU-19.82",
    "math.GR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-22",
  "publication_date": "2026-10-22",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-22",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/kou-19-82/",
  "pdf_url": "https://eulersolve.org/papers/kou-19-82/paper.pdf?v=d06e29630a6a",
  "doi": "10.5281/zenodo.23548923",
  "zenodo_record_url": "https://zenodo.org/records/23548923",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Excluded: PSL(2,q)-factors with q odd, q = +-1 mod 8, whose induced group contains PGL(2,q) (a limit of the method, not a counterexample); A_n for n >= 7, other groups of Lie type and the sporadic groups are not treated.",
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}