{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R595",
  "content_hash": null,
  "slug": "pp12-claim-collineation-orders",
  "type": "claim",
  "title": "Every collineation group has order 1, 2, or 3",
  "summary": "The strongest published symmetry restriction leaves only trivial, involutory, or order-three collineation groups.",
  "relevance": "For A projective plane of order 12, record pp12-claim-collineation-orders (“Every collineation group has order 1, 2, or 3”) records a bound, answer, status fact, or structural consequence. The record states: The strongest published symmetry restriction leaves only trivial, involutory, or order-three collineation groups.",
  "relevance_source": "recorded",
  "body": "Akiyama, Suetake, and Tanaka prove that a projective plane of order 12 cannot admit a collineation group of order 4. Their paper combines this result with the earlier exclusions and states the resulting classification:\n\\[\n|G|\\in\\{1,2,3\\}\n\\]\nfor every collineation group \\(G\\) of a hypothetical plane of order 12. In particular, its full collineation group would have one of these three orders.\n\nThis theorem sharply limits symmetry-based enumeration. A search restricted to automorphism groups of order at least 4 covers an empty class. A complete computation must still handle planes with a trivial group or a group of order 2 or 3.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "the full class of projective planes of order 12 and every collineation group acting on any such plane"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1002/jcd.21869",
      "locator": "Kenzi Akiyama, Chihiro Suetake, and Masaki Tanaka, Projective planes of order 12 do not have a collineation group of order 4, Journal of Combinatorial Designs 31(2) (2023), 87-123, abstract and main theorem"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1002/jcd.21869",
    "locator": "Kenzi Akiyama, Chihiro Suetake, and Masaki Tanaka, Projective planes of order 12 do not have a collineation group of order 4, Journal of Combinatorial Designs 31(2) (2023), 87-123, abstract and main theorem"
  },
  "models": [],
  "relations": [
    {
      "slug": "R597",
      "title": "Existence at order 12 remains open",
      "object_type": "claim",
      "relation": "constrains",
      "direction": "outgoing"
    },
    {
      "slug": "projective-plane-order-12",
      "title": "projective plane order 12",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}