{
  "schema_version": 1,
  "problem_number": "KOU-19.93",
  "title": "On Part b) of Kourovka Notebook Problem 19.93: Finite p-Groups with Unbounded Lower p-Central Factors and No C_p ≀ C_p Quotient for p = 2, 3, 5",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Problem 19.93 of the Kourovka Notebook, posed by P. Spiga in 2018, conjectures that there is a function f: ℕ × ℕ → ℕ such that every factor of the lower p-central series of a finite p-group with d generators and without an epimorphic image isomorphic to C_p ≀ C_p has order at most f(p,d); part a) concerns p ≥ 7, part b) p = 2, 3, 5. M. Kinyon and J. Urban refuted part a), in a preprint that declares AI assistance, with a filtered Golod–Shafarevich inequality and a counting bound for truncated group algebras; the method of this note is theirs. We show that part b) is false: for each of p = 2, 3, 5 no bound f(p,d) exists (for p = 5 with any d ≥ 3, for p = 3 with any d ≥ 5, for p = 2 with any d ≥ 7). What is new is the choice of relators: x_i^p and (x_i x_j)^p on three generators for p = 5 and on five for p = 3, where the argument of Kinyon and Urban in C_p ≀ C_p, applied to each pair of generators, excludes this quotient; and twelve products of squares and commutators on seven generators for p = 2, where the exclusion of C_2 ≀ C_2 becomes a finite statement about twelve bilinear forms over 𝔽_2, checked exactly in two ways. We also observe why the Golod–Shafarevich argument with the relations used for p ≥ 7 fails for p = 2, 3, 5. Every step is proved in the note, including the Golod–Shafarevich inequality in the form used. With part a), the conjecture fails for every prime. The cases (p,d) = (5,2), (3,2), (3,3), (3,4) and (2,2), …, (2,6) are not decided. No printed answer to part b) was found in the sources searched. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.GR"
  ],
  "keywords": [
    "finite p-groups",
    "lower p-central series",
    "lower exponent-p central series",
    "wreath products",
    "Golod–Shafarevich inequality",
    "filtered algebras",
    "truncated group algebras",
    "counterexample",
    "Kourovka Notebook",
    "Spiga's conjecture",
    "UnsolvedMath",
    "KOU-19.93",
    "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-19-93/",
  "pdf_url": "https://eulersolve.org/papers/kou-19-93/paper.pdf?v=d662832384f6",
  "doi": "10.5281/zenodo.23544457",
  "zenodo_record_url": "https://zenodo.org/records/23544457",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The method is that of Kinyon and Urban, who refuted part a); the cases (p, d) = (5, 2), (3, 2), (3, 3), (3, 4) and (2, 2) to (2, 6) are not decided.",
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    "source.zip": {
      "sha256": "51177c6b3cf30b4448fda17a78a12c0729df37e418bcef56686c6701499278ad"
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    "verification_report.md": {
      "sha256": "d48e8a494679f89e2a94c8a891a28cf4f50ac37d928da6b935749ad8c3c893c8"
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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."
}