{
  "schema_version": 1,
  "problem_number": "KOU-15.67",
  "title": "The Chevalley Group F4(Z) Is Generated by Three Involutions, Two of Which Commute",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In Problem 15.67 of the Kourovka Notebook (2002), Ya. N. Nuzhin asked which adjoint Chevalley groups of normal type over the integers are generated by three involutions, two of which commute. By work of Tamburini and Zucca, Nuzhin, Timofeenko, and Kazakova and Nuzhin, the answer was known for the types A_l, B_2 = C_2, G_2, E_6, E_7 and E_8; the types B_l, C_l (l ≥ 3), D_l (l ≥ 4) and F_4 were open. Our result is a partial answer: the adjoint Chevalley group F_4(Z) is generated by three involutions two of which commute (computer-assisted proof with an exact certificate); the types B_l, C_l (l ≥ 3) and D_l (l ≥ 4) stay open. The involutions are explicit and follow the pattern of Timofeenko's triples for E_l: b and c lift the two halves of the bipartite Coxeter element, and a is a product of commuting root elements and a torus element (four root elements of both lengths here, two in Timofeenko's triples). A direct argument shows that they are involutions and reduces ab = ba to four signs of structure constants. Generation is proved by a straight-line program with 1064 nodes which produces the 48 root elements x_r(1); a checker that uses only the Python standard library verifies it exactly in a few seconds, and three independent programs confirm it. The formulas refer to a Chevalley basis that we describe by its extraspecial signs; the existence statement does not depend on this choice. As consequences, the reductions modulo every prime p generate F_4(p) in the same way, and the least number of generating involutions of F_4(Z) with product 1 is 5. No printed answer for F_4 was found in the sources searched (zbMATH, arXiv, OpenAlex, Crossref, Semantic Scholar, web searches); a search that finds nothing is not a proof of novelty. This is an unrefereed note.",
  "result_type": "PARTIAL_RESULT",
  "categories": [
    "math.GR"
  ],
  "keywords": [
    "Chevalley groups over the integers",
    "exceptional group F4",
    "generation by involutions",
    "three involutions two of which commute",
    "Chevalley basis",
    "extraspecial pairs",
    "straight-line program",
    "computer-assisted proof",
    "exact certificate",
    "partial answer",
    "Kourovka Notebook",
    "UnsolvedMath",
    "KOU-15.67",
    "math.GR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/kou-15-67/",
  "pdf_url": "https://eulersolve.org/papers/kou-15-67/paper.pdf?v=2c9f011dc50c",
  "doi": "10.5281/zenodo.23585662",
  "zenodo_record_url": "https://zenodo.org/records/23585662",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The types B_l, C_l (l ≥ 3) and D_l (l ≥ 4) stay open; the proof is computer-assisted (exact certificate) and unrefereed.",
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    "source.zip": {
      "sha256": "cd4ab5807191712e0b7caa0fadf34dc2a1199a2df61de388df8c24155ccdcfdb"
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    "verification_report.md": {
      "sha256": "ba5fe0aa1e1b1ac84ab6f622b8464109c43ee5646c1c4b491384ae62ea56615e"
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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."
}