{
  "schema_version": 1,
  "problem_number": "KOU-15.75",
  "title": "On Plotkin's Question on Two-Variable Identities Characterizing Finite Soluble Groups: Two Infinite Families of PSL(2,p) and Finite Certificates",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "B. I. Plotkin asked in the Kourovka Notebook (Problem 15.75 b), 2002) whether a finite group is soluble if and only if it satisfies one of the identities u_n = 1, where u_1 = [x,y] and u_{n+1} = [[u_n,x],[u_n,y]]. We give a partial answer: the question reduces to the minimal simple groups; PSL(2,p) is settled for all primes p ≡ ±1 (mod 16) and, in the Notebook's commutator convention, for all primes p ≡ 0, ±1 (mod 7), and every minimal simple group of order less than |PSL(2,10007)| has a pair whose sequence never reaches 1; the question stays open. The reduction to Thompson's list of minimal simple groups is standard: it was outlined for this sequence by Nikolova and Plotkin (2001) and carried out by Bandman, Greuel, Grunewald, Kunyavskii, Pfister and Plotkin for another sequence. The families are explicit: if x, y ∈ SL(2,p), tr x = 0 and tr y = tr xy is a root of 2t^4 − 4t^2 + 1, then u_3 = −u_2 ≠ ±1 (in the other commutator convention u_4 = u_3 ≠ ±1); if tr x = 1 and tr y = tr xy is a root of t^3 ± t^2 − 2t ∓ 1, then u_4 = u_2 ≠ ±1 (in the Notebook's convention). By Dirichlet's theorem they contain half of the primes p > 3 with p ≡ ±2 (mod 5), for which PSL(2,p) is minimal simple. Checked finite certificates, pairs with u_M = u_{M+L} ≠ 1, settle PSL(3,3), the 626 groups PSL(2,q) of Thompson's list with q ≤ 10^4, and Sz(8), Sz(32), Sz(128); for PSL(3,3) and PSL(2,p), 5 ≤ p < 100, Nikolova and Plotkin had reported such pairs, found by computer. Not decided are the other groups of Thompson's list: PSL(2,p) for primes p > 10^4 outside the families, PSL(2,2^r) for primes r ≥ 17, and PSL(2,3^r) and Sz(2^r) for primes r ≥ 11. This is computational progress and two infinite families, not a near-solution. No printed answer to the question was found in the sources searched; we claim no priority. This is an unrefereed note.",
  "result_type": "PARTIAL_RESULT",
  "categories": [
    "math.GR"
  ],
  "keywords": [
    "Plotkin",
    "two-variable identities",
    "Engel-like sequences",
    "finite soluble groups",
    "minimal simple groups",
    "PSL(2,p)",
    "Suzuki groups",
    "trace map",
    "partial answer",
    "Kourovka Notebook problems",
    "UnsolvedMath",
    "KOU-15.75",
    "math.GR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-22",
  "publication_date": "2026-10-22",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-22",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/kou-15-75/",
  "pdf_url": "https://eulersolve.org/papers/kou-15-75/paper.pdf?v=bfeb29c3cf3c",
  "doi": "10.5281/zenodo.23552969",
  "zenodo_record_url": "https://zenodo.org/records/23552969",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The question stays open (PSL(2,p) outside the two families for p above 10^4, PSL(2,2^r), PSL(2,3^r) and Sz(2^r) for larger r); Nikolova and Plotkin (2001) had outlined the reduction and found pairs for small groups.",
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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."
}