{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-quotient-group",
  "semantic_family_id": "sf-finite-quotient-group",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限商群"
  },
  "summary": {
    "zh-Hant": "有限群G依正規子群N的陪集形成商群，元素是cosets且乘法(gN)(hN)=(gh)N必須與代表元無關。"
  },
  "statement": {
    "zh-Hant": "N須先由完整表認證為normal subgroup。全部left cosets形成disjoint complete partition，quotient elements是coset sets而非代表元。乘法表以coset index記錄，且對每一對cosets及所有代表元替換完整驗product落入同一coset。identity為N，inverse由g^{-1}N給出。若N不正規，明示代表元選擇可產生不同product cosets，因此不能建商群。商群不自動commutative，也不提前宣稱第一同構定理。"
  },
  "formula": {
    "tex": "Q(G,N)=C",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>Q</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>N</mi><mo>)</mo></mrow><mo>=</mo><mi>C</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "Q",
          "glyph": "Q"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "G",
            "glyph": "G"
          },
          {
            "type": "symbol",
            "name": "N",
            "glyph": "N"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "e412023b88df7b7cd6ed9a23b986eedd30b2176c8eb3ffb6d3a438abc95d90ad"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題族使用完整有限Cayley table、identity與明示nonempty subset。"
    },
    {
      "id": "a2",
      "zh-Hant": "coset partition與所有代表元pairs均完整枚舉。"
    }
  ],
  "symbols": [
    {
      "token": "Q",
      "role": "quotient_group_operator",
      "role_zh": "商群構造",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "finite_group",
      "role_zh": "有限群",
      "scope": "representative_formula"
    },
    {
      "token": "N",
      "role": "normal_subgroup",
      "role_zh": "正規子群",
      "scope": "representative_formula"
    },
    {
      "token": "C",
      "role": "coset_carrier",
      "role_zh": "陪集載體",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-normal-subgroup",
      "title_zh": "有限正規子群",
      "reason_zh": "well-definedness必要條件。"
    },
    {
      "id": "mko-quotient-set",
      "title_zh": "商集合",
      "reason_zh": "coset equivalence classes。"
    },
    {
      "id": "mko-finite-group",
      "title_zh": "有限群",
      "reason_zh": "群乘法與inverse。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "先建立完整coset partition。",
      "商乘法需對所有代表元替換不變。",
      "nonnormal witness直接否證well-definedness。"
    ],
    "common_misconception_zh": "把單一representative當quotient element，或未驗normal便直接乘cosets。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只建立有限商群；不自動聲稱commutative或與image同構。",
    "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": "加入有限群第一同構定理。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-quotient-group.md 整理；2026-10-15 脈衝 R02（使用者手動接續 2026-11-14 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}