{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-normal-subgroup",
  "semantic_family_id": "sf-finite-normal-subgroup",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限正規子群"
  },
  "summary": {
    "zh-Hant": "有限群G的子群H若對每個g皆滿足gHg^{-1}=H，則H為正規子群；等價地每個左陪集等於相應右陪集。"
  },
  "statement": {
    "zh-Hant": "G須由完整Cayley table認證，H須先完整通過identity、product與inverse closure。正規性要求對每個g∈G完整驗gHg^{-1}=H；等價條件是每個g皆有gH=Hg。一個明示g,h使ghg^{-1}不屬H即可否證normal。trivial subgroup與whole group均正規。有限性本身、抽樣共軛或只檢生成元但未證生成責任都不足；本節點尚不宣稱quotient multiplication well-defined。"
  },
  "formula": {
    "tex": "N(H,G)=1",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>N</mi><mo>(</mo><mi>H</mi><mo>,</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "N",
          "glyph": "N"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "H",
            "glyph": "H"
          },
          {
            "type": "symbol",
            "name": "G",
            "glyph": "G"
          }
        ]
      },
      "rhs": {
        "type": "number",
        "value": 1
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "9168f31e65e42667f725c4c03874342d3e372d3ef77270bce6f2c93e644c0d3d"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題族使用完整有限Cayley table、identity與明示subset，不接受sampled multiplication。"
    },
    {
      "id": "a2",
      "zh-Hant": "conjugate sets、left/right cosets與inverse均由完整表精確枚舉。"
    }
  ],
  "symbols": [
    {
      "token": "N",
      "role": "normality_predicate",
      "role_zh": "正規性述詞",
      "scope": "representative_formula"
    },
    {
      "token": "H",
      "role": "candidate_subgroup",
      "role_zh": "候選子群",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "finite_group",
      "role_zh": "有限群",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-group",
      "title_zh": "有限群",
      "reason_zh": "完整Cayley table。"
    },
    {
      "id": "mko-subgroup-homomorphism",
      "title_zh": "子群與群同態",
      "reason_zh": "subgroup closure。"
    },
    {
      "id": "mko-quantifier",
      "title_zh": "量詞",
      "reason_zh": "對每個g的全稱責任。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "subgroup資格與normal資格是兩層不同義務。",
      "完整共軛集合相等與逐g左右陪集相等是等價核對。",
      "單一逃出H的conjugate element可否證normal。"
    ],
    "common_misconception_zh": "因群有限或H大小整除|G|便稱H正規，或只抽查幾個g。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪限完整有限群模型；尚未物化商群運算或第一同構定理。",
    "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": "加入有限商群、kernel與第一同構定理。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-finite-normal-subgroup.md 整理；2026-10-15 脈衝 R01（使用者手動接續 2026-11-13 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"
  }
}