{
  "schema_version": 1,
  "problem_number": "KOU-7.27",
  "title": "Levchuk's Problem on Diagonal Matrices in SL_n(q) (Kourovka Notebook 7.27): An Affirmative Answer as a Corollary of Theorems of Fulman and Guralnick",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "V. M. Levchuk asked in 1980 (Kourovka Notebook, Problem 7.27) whether, for all sufficiently large q, the group SL_n(q) contains a diagonal matrix lying in no proper irreducible subgroup except block-monomial ones. We show that the answer is yes for every n, as a short corollary of density theorems of Fulman and Guralnick; it is not presented as a new theorem. For fixed n and q → ∞, almost all regular diagonal matrices d of SL_n(q) have the property that every proper subgroup containing d that is irreducible over F_q permutes the summands of a decomposition of F_q^n into coordinate subspaces. The deduction combines the density theorems with an elementary count of the classes of regular diagonal matrices, a description of the maximal subgroups of maximal rank, and an eigenvalue argument excluding extension field subgroups; the bound for q is not explicit. For n = 2, Dickson's list of subgroups gives the complete answer: a diagonal matrix with this property exists if and only if q ∈ {4, 8} or q ≥ 13 (if irreducibility is meant absolutely, also for q = 2), and diag(w, w⁻¹) works for every generator w of F_q^* when q ≥ 13. No printed statement of the general answer was found, and no priority is claimed. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GR"
  ],
  "keywords": [
    "special linear group",
    "diagonal matrices",
    "regular semisimple elements",
    "irreducible subgroups",
    "block-monomial subgroups",
    "imprimitive subgroups",
    "maximal subgroups of maximal rank",
    "Aschbacher classes",
    "Fulman–Guralnick density theorems",
    "Dickson's theorem",
    "Kourovka Notebook",
    "Problem 7.27",
    "Levchuk",
    "complete answer (corollary of published theorems)",
    "UnsolvedMath",
    "KOU-7.27",
    "math.GR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-18",
  "publication_date": "2026-10-18",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-18",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/kou-7-27/",
  "pdf_url": "https://eulersolve.org/papers/kou-7-27/paper.pdf?v=852485fc8d86",
  "doi": "10.5281/zenodo.23445980",
  "zenodo_record_url": "https://zenodo.org/records/23445980",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The answer follows from published density theorems of Fulman and Guralnick; the note writes out the deduction and settles n = 2 completely. No priority is claimed. Unrefereed note.",
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    "source.zip": {
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    "verification_report.md": {
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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."
}