{
  "schema_version": 1,
  "problem_number": "KOU-15.98",
  "title": "Minimal Non-F-Groups with Sylow Residual and the Unitary Groups PSU(3, p^{2n}): A Negative Answer to Kourovka Notebook Problem 15.98",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In Problem 15.98 of the Kourovka Notebook (2002), L. A. Shemetkov asked whether, for a saturated formation F and a finite soluble minimal non-F-group G whose F-residual is a Sylow p-subgroup, this residual is always isomorphic to a factor group of the Sylow p-subgroup of PSU(3, p^{2n}). For the formation of nilpotent groups this is a theorem of Mazurov and Syskin (p = 2) and of Zhurtov and Syskin on Schmidt groups. We show that the answer is no: there is a saturated formation F and a finite soluble minimal non-F-group G of order 2304 whose F-residual is a Sylow 2-subgroup isomorphic to no factor group of a Sylow 2-subgroup of PSU(3, 2^{2n}); 2304 is the least order of such a counterexample. The obstruction is elementary: in a nonabelian factor group Q of the Sylow p-subgroup of PSU(3, p^{2n}) the commutators [y, x], x ∈ Q, fill Q' for every non-central y, while our residual, a special group of order 2^8 with centre of order 4, has a non-central element with only two commutators. The example comes from an elementary construction, a cyclic group acting on a special p-group, which gives a second counterexample of order 3^8·2^4 and shows that the answer stays no for subgroup-closed formations containing all supersoluble groups, for formations contained in N_p N_r, and for the formations of metanilpotent and of nilpotent-by-abelian groups. For the formation of supersoluble groups the answer is yes; we include a proof, not claimed as new, which uses the theorem of Zhurtov and Syskin as stated in its review. The least-order statement does not use that theorem. No printed answer to the question was found in the sources we searched; a search that finds nothing is not a proof of novelty, and the construction may be known to specialists. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.GR"
  ],
  "keywords": [
    "saturated formation",
    "local formation",
    "minimal non-F-group",
    "F-critical group",
    "F-residual",
    "Schmidt groups",
    "Sylow subgroups of PSU(3, q^2)",
    "semi-extraspecial groups",
    "special p-groups",
    "supersoluble groups",
    "counterexample",
    "Kourovka Notebook",
    "UnsolvedMath",
    "KOU-15.98",
    "math.GR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-21",
  "publication_date": "2026-10-21",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-21",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/kou-15-98/",
  "pdf_url": "https://eulersolve.org/papers/kou-15-98/paper.pdf?v=0d8be9b44f1a",
  "doi": "10.5281/zenodo.23544530",
  "zenodo_record_url": "https://zenodo.org/records/23544530",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For supersoluble groups the answer is yes (proof included, not claimed new; it uses the Zhurtov-Syskin theorem as stated in its review); the construction may be known to specialists.",
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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."
}