{
  "schema_version": 1,
  "problem_number": "KOU-21.68",
  "title": "A Semi-Abelian Group of Order 768 That Is Not an M-Group",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "M. Kida conjectured that every finite semi-abelian group is monomial (J. Group Theory, 2025; Kourovka Notebook, Problem 21.68). We show that the conjecture is false. If a finite group W acts on a finite abelian group N, a linear character of N has stabiliser T in W, and T has an irreducible character that is not monomial, then N ⋊ W is not an M-group. We apply this to the augmentation submodule of the permutation module F₂[W/T], where W = E ⋊ A₄ is the index-two subgroup of C₂ ≀ A₄ and T ≅ SL(2,3) is the binary tetrahedral group acting on the eight quaternion units. The result is a semi-abelian group of order 768 = 2⁸·3 with a non-monomial irreducible character of degree 8. The proof uses only Clifford theory. The example was also checked by exact computation, including every subgroup of index 8. This is an unrefereed note; minimality of the order is not claimed.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.GR",
    "math.RT"
  ],
  "keywords": [
    "semi-abelian groups",
    "M-groups",
    "monomial groups",
    "Clifford theory",
    "character theory",
    "Kourovka Notebook",
    "counterexamples",
    "KOU-21.68",
    "open mathematics",
    "mathematical proof",
    "math.GR",
    "math.RT"
  ],
  "manuscript_version_date": "2026-09-27",
  "publication_date": "2026-09-27",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-27",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/kou-21-68/",
  "pdf_url": "https://eulersolve.org/papers/kou-21-68/paper.pdf?v=cac6ab161f3b",
  "doi": "10.5281/zenodo.23000305",
  "zenodo_record_url": "https://zenodo.org/records/23000305",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": null,
  "files": {
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    "source.zip": {
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    "verification_report.md": {
      "sha256": "e3d344e6bdc00ddffc15c16a14663d644eb85846dabe590aa590e763ebbc2431"
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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."
}