{
  "schema_version": 1,
  "problem_number": "OWR-11129-006",
  "title": "Elements Powering to Strong Involutions in Finite Classical Groups of Odd Characteristic: Order 1/log n and a Log-Periodic Asymptotic",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let G be a finite n-dimensional classical group over a field of odd order q, and let Q be the set of elements g of even order whose involution g^(|g|/2) has a fixed space of dimension in [n/3, 2n/3). Lübeck, Niemeyer and Praeger proved |Q|/|G| ≥ c/log n, and Praeger, with Lübeck and Niemeyer, asked at Oberwolfach in 2011 for the asymptotic value of |Q|/|G|, in particular whether |Q|/|G| = O(1/log n) (printed as O(log n), evidently a misprint). For the groups X of the table of Lübeck, Niemeyer and Praeger and for Sp_2m(q), SO_2m+1(q) and SO±_2m(q) we prove |Q|/|X| ≤ C/log n with absolute constants (C = 100 for GL_n(q) and GU_n(q); 45 for Sp_2m(q), GSp_2m(q), SO_2m+1(q), GO_2m+1(q); 80 for SO±_2m(q), GO±_2m(q)^0), uniformly in q; with their lower bound, |Q|/|H| = Θ(1/log n) for every H between the groups S ≤ X of a line of the table, the upper constant carrying a factor at most q + 1 for the other H, among them SL_n(q) and SU_n(q). For the eight groups listed we also prove (log n)|Q|/|X| = Φ(log₂ log n) + o(1) uniformly in q, where Φ(u) = (log 2/3) Σ_{k∈Z} 2^(u−k) exp(−2^(u−k)/3) is 1-periodic with mean 1, not constant, and within 10⁻⁵ of 1. Thus |Q|/|X| is 1/log n up to a factor that tends to within 10⁻⁵ of 1, while (log n)|Q|/|X| has no limit. The set Q depends only on semisimple parts, and the torus formula of Niemeyer and Praeger reduces the problem to random permutations, signed for the symplectic and orthogonal groups, whose cycles carry independent 2-adic levels; what is added here is a level-by-level upper bound, the torus model for the groups of similitudes, and the asymptotic analysis of the levels. The full orthogonal groups O±_2m(q) are not covered; for SL_n(q), SU_n(q) and the groups strictly between S and X only the order of magnitude is proved, and for Ω only the upper bound; the convergence is slow, and the numerical values are evidence only. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GR",
    "math.PR",
    "math.CO"
  ],
  "keywords": [
    "finite classical groups",
    "strong involutions",
    "odd characteristic",
    "proportions of elements",
    "maximal tori",
    "quokka sets",
    "random permutations",
    "cycle index",
    "2-adic valuation",
    "log-periodic asymptotics",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-11129-006",
    "math.GR",
    "math.PR",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-19",
  "publication_date": "2026-10-19",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-19",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-11129-006/",
  "pdf_url": "https://eulersolve.org/papers/owr-11129-006/paper.pdf?v=0efc3811754f",
  "doi": "10.5281/zenodo.23483629",
  "zenodo_record_url": "https://zenodo.org/records/23483629",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Proved for the groups of the Luebeck-Niemeyer-Praeger table; the full orthogonal groups O^(+-)_2m(q) are not covered, and for SL_n, SU_n and the intermediate groups only the order of magnitude is proved.",
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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."
}