{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-group",
  "semantic_family_id": "sf-finite-group",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限群"
  },
  "summary": {
    "zh-Hant": "有限群由有限集合與完整封閉Cayley表構成，並滿足結合律、唯一單位元及每元素逆元。"
  },
  "statement": {
    "zh-Hant": "有限集合G配備二元運算*。若運算對G封閉、對所有三元組結合、存在雙側單位元且每元素有雙側逆元，則(G,*)為有限群。完整Cayley表可窮舉核對指定有限模型。"
  },
  "formula": {
    "tex": "G(S)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>G</mi><mo>(</mo><mi>S</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "G",
        "glyph": "G"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "S",
          "glyph": "S"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "1cf1e8405feabea2c0bc49064a7796ef19f3f4e8a8d4f16a18190920f471436f"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "元素以0到n-1標記，題目提供n×n完整運算表。"
    },
    {
      "id": "a2",
      "zh-Hant": "抽樣列或少數三元組不證群；結合律必須窮舉全部n³三元組。"
    }
  ],
  "symbols": [
    {
      "token": "G",
      "role": "finite_group_candidate",
      "role_zh": "有限群候選",
      "scope": "representative_formula"
    },
    {
      "token": "S",
      "role": "carrier_set",
      "role_zh": "有限載體",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-set",
      "title_zh": "集合",
      "reason_zh": "群有固定載體集合。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "二元運算必須總且單值。"
    },
    {
      "id": "mko-relation",
      "title_zh": "二元關係與集合投影",
      "reason_zh": "Cayley表以輸入對指定輸出。"
    },
    {
      "id": "mko-proof",
      "title_zh": "證明",
      "reason_zh": "群資格需封閉全部公理責任。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "封閉表示表內每格輸出仍是G元素。",
      "結合律比較(a*b)*c與a*(b*c)的全部三元組，不等同交換律。",
      "單位元需左右都成立；逆元也需左右乘都得到同一單位元。"
    ],
    "common_misconception_zh": "只看表的對稱性或幾個案例就宣稱是群，或把單側單位／逆元當雙側。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "完整有限表可精確核對指定候選；不外推無限群或未列運算。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入子群、同態、陪集、正規子群與商群。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-group.md 整理；2026-10-13 脈衝 R05（使用者手動接續 2026-10-22 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}