{
  "schema_version": 1,
  "problem_number": "OWR-793-006",
  "title": "A Negative Answer to a Question of Praeger and Seress on Minimal Subdegrees of Primitive Groups of Twisted Wreath Type",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let G be a finite primitive permutation group of twisted wreath type, with socle T^k and point stabiliser P, and let H be its primitive component: the group induced on one coordinate of the product decomposition Δ^k, where |Δ| = |T|. Praeger and Seress (Oberwolfach Report 12/2005, on joint work with Giudici, Li and Trofimov) state that the least non-trivial subdegree satisfies MinSubDeg(G) ≤ k · MinSubDeg(H), and they ask in their Question 4 (p. 692) whether this inequality is always strict. We give a negative answer to Question 4. Let P = PGL(3,4) act on the 21 points of the projective plane of order 4, let Q be the stabiliser of a point y, let T = PSL(2,4) ≅ A_5, and let Q act on T through its action on the quotient space F_4^3 / y. The twisted wreath product G = T twr P, acting on its base group, is primitive of twisted wreath type of degree 60^21, with k = 21, H = A_5 × A_5, MinSubDeg(H) = 12 and MinSubDeg(G) = 252 = 21 · 12. By a reduction which is known in substance, and which we prove again, the last equality is a statement about the subgroups of PGL(3,4) of order greater than 240. We prove this statement in two ways: computer-assisted, by exhaustive exact computations with independently written programs; and by a written argument from the list of the maximal subgroups of PSL(3,4) and PGL(3,4) in the tables of Connor and Leemans. Computer-assisted, we also find equality for the top groups PΓL(3,4) (315 = 21 · 15) and PSL(3,5) (465 = 31 · 15), where H is the full holomorph of A_5, and strict inequality for PSL(3,4) and PΣL(3,4). The full paper of Giudici, Li, Praeger, Seress and Trofimov (2006) and its companion paper were not accessible to us; we make no statement that depends on their content. No answer to Question 4 was found in the literature accessible to us; a search that finds nothing is not a proof of novelty. We claim neither that the example is the smallest one nor a classification of the cases of equality, and the question remains open under additional hypotheses. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "keywords": [
    "primitive permutation groups",
    "twisted wreath products",
    "O'Nan–Scott type TW",
    "minimal subdegree",
    "suborbits",
    "subdegrees",
    "PGL(3,4)",
    "projective plane of order 4",
    "computer-assisted proof",
    "negative answer",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-793-006",
    "math.GR",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-11",
  "publication_date": "2026-10-11",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-11",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-793-006/",
  "pdf_url": "https://eulersolve.org/papers/owr-793-006/paper.pdf?v=f0336d4a9f89",
  "doi": "10.5281/zenodo.23299865",
  "zenodo_record_url": "https://zenodo.org/records/23299865",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Question 4 of C. E. Praeger and Á. Seress (Oberwolfach Report 12/2005, p. 692) in the negative: the twisted wreath product of A_5 by PGL(3,4) attains the upper bound. The statement on the subgroups of PGL(3,4) is proved computer-assisted and by a written argument from the maximal subgroups in the tables of Connor and Leemans; the further examples are computer-assisted. The full paper of Giudici, Li, Praeger, Seress and Trofimov (2006) and its companion paper were not accessible to us; no statement depends on their content, and novelty is not certified. The reduction lemmas are known in substance. The question remains open under additional hypotheses.",
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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."
}