{
  "schema_version": 1,
  "problem_number": "KOU-20.52",
  "title": "Groups of Odd Order with Conjugate Deficiency 16m (Kourovka Notebook Problem 20.52)",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a finite group G let k(G) be its number of conjugacy classes and r(G) = |G| − k(G). Burnside proved that 16 divides r(G) when |G| is odd, and D. MacHale asked (Kourovka Notebook, Problem 20.52) whether every positive multiple 16m is r(G) for a non-abelian group G of odd order. Since r(G × C_c) = c·r(G), the question reduces to m = 2^j; this is Problem 5 of Buckley and MacHale (2013), who settled j ≤ 8. We give explicit groups for every j ≤ 13, so every m below 16384 occurs, and, by a computer-assisted search with exact certificates, for every j ≤ 120 except j ∈ {14, 34, 64}. Besides direct products with Frobenius groups, the new examples are semidirect products (M₁ × M₂) ⋊ H of abelian groups by groups H of order at most 2187, mostly 3-groups, acting through two cyclic quotients; their class numbers are given by a closed formula that we prove: if H acts on Mᵢ through H/Kᵢ, every element outside Kᵢ acting without non-trivial fixed points, then k(G) = Σ_S ∏_{i∈S} (|Mᵢ| − 1)·k(K_S)/|H : K_S| with K_S = ⋂_{i∈S} Kᵢ. For example (C₁₃ × C₃₃₇) ⋊ 3^{1+4}₊ has r(G) = 2²⁰, which gives j = 16. For j = 14, that is m = 16384, we show that no group in several natural families works, among them P × F for every group P of odd order and every Frobenius group F with abelian kernel and complement of prime order. The question remains open, first for m = 2¹⁴. This is an unrefereed note.",
  "result_type": "PARTIAL_RESULT",
  "categories": [
    "math.GR"
  ],
  "keywords": [
    "Kourovka Notebook",
    "MacHale",
    "Burnside",
    "conjugacy classes",
    "class number",
    "groups of odd order",
    "conjugate deficiency",
    "Frobenius groups",
    "Clifford theory",
    "computer-assisted proof",
    "partial answer",
    "UnsolvedMath",
    "KOU-20.52",
    "math.GR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-27",
  "publication_date": "2026-10-27",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-27",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/kou-20-52/",
  "pdf_url": "https://eulersolve.org/papers/kou-20-52/paper.pdf?v=9eb72ad6f25c",
  "doi": "10.5281/zenodo.23685552",
  "zenodo_record_url": "https://zenodo.org/records/23685552",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The question stays open, first for m = 2^14; the exponents from 15 to 120 rest on computer-assisted certificates (Python, GAP 4.16.1) with exact checks and Pocklington primality proofs; unrefereed.",
  "files": {
    "paper.pdf": {
      "sha256": "9eb72ad6f25cbfae6cf5bf01c179024cc6ef1e415a2b8ceed01f4641226ca802"
    },
    "source.zip": {
      "sha256": "f257098f3c6f3f1d08881d0d1bb19b23d46abd8f9486ee5ee29b0c1f463b84d9"
    },
    "verification_report.md": {
      "sha256": "a6868fc29a6ff9a73172dbae574e13f23c319a76162717604e80e24cc1358f8d"
    }
  },
  "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."
}