{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1364",
  "content_hash": null,
  "slug": "tarski-monster-exponent-five-status-20260801",
  "type": "claim",
  "title": "Current checked status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-27. The MathOverflow page has no answers and its comments connect the case to the small odd-exponent Burnside problem. The checked modern theorem proves infinitude for much larger odd exponents and does not reach exponent five.",
  "relevance": "Records the strongest checked neighboring results and the exact remainder future work must settle.",
  "relevance_source": "recorded",
  "body": "A dated independent review on 2026-08-01 checked the structured sources below, the complete visible source discussion, exact-title and equivalent-formulation searches, and the current TheoremDB corpus. On 2026-07-27 the unanswered MathOverflow question and all comments were checked. No construction or impossibility proof is posted. Olshanskii's Tarski-monster constructions work for sufficiently large primes. The exact exponent five lies far below the range supplied by those methods. Atabekyan and Ivanov, arXiv:2303.15997, prove infinitude of free Burnside groups for odd exponents at least 557. Their quantitative threshold leaves exponent five untouched. The restricted Burnside group of exponent five is finite, while the ordinary free Burnside group \\(B(2,5)\\) has long-standing unresolved behavior in the sources checked. A Tarski monster would in particular provide an infinite exponent-five group. Trap: an infinite group generated by elements of order five need not have exponent five, and an infinite exponent-five group need not have the Tarski subgroup property.\n\nA complete resolution must satisfy: For a positive answer, construct a group \\(G\\) and prove infinitude, exponent five, and the assertion that every nontrivial proper subgroup is cyclic of order five. For a negative answer, prove that no infinite exponent-five group can have the stated subgroup structure.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/138368/tarski-monster-group-with-prime-5",
      "locator": "Dataset references and independent 2026-08-01 status search."
    },
    "missing": [
      "source",
      "command",
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    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/138368/tarski-monster-group-with-prime-5",
    "locator": "Dataset references and independent 2026-08-01 status search."
  },
  "models": [],
  "relations": [
    {
      "slug": "R1363",
      "title": "Complete the stated acceptance conditions",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "R1772",
      "title": "Dated status and exact unresolved remainder",
      "object_type": "claim",
      "relation": "supersedes",
      "direction": "incoming",
      "metadata": {
        "reason": "Replaces unreadable status prose with the dated review from 2026-08-01."
      }
    },
    {
      "slug": "tarski-monster-exponent-five",
      "title": "tarski monster exponent five",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}