{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-first-isomorphism-theorem-finite-groups",
  "semantic_family_id": "sf-first-isomorphism-theorem-finite-groups",
  "type": "theorem",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限群第一同構定理"
  },
  "summary": {
    "zh-Hant": "有限群同態f:G→Q由kernel cosets誘導出G/ker f到im f的群同構，而非自動同構到整個codomain。"
  },
  "statement": {
    "zh-Hant": "來源與目標均須以完整Cayley table認證，f為complete total homomorphism；kernel、image與quotient cosets先完整建立。誘導映射bar f(g ker f)=f(g)必須對每個coset的所有representatives給同值，並完整驗乘法保存、injective與surjective-to-image。若f不surjective，結論仍僅為G/ker f≅im f，不得把image換成整個codomain。有限枚舉只證指定模型；非同態不得套用定理。"
  },
  "formula": {
    "tex": "I(f)=h",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>I</mi><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>=</mo><mi>h</mi></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "I",
          "glyph": "I"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "f",
            "glyph": "f"
          }
        ]
      },
      "rhs": {
        "type": "symbol",
        "name": "h",
        "glyph": "h"
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "5785d2166ab8235b40c34a47db700f3938fd6b2fb72ad8e8f84dc2903d32687c"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題族使用兩個完整有限群表與完整total mapping array。"
    },
    {
      "id": "a2",
      "zh-Hant": "quotient carrier與image carrier均由完整有限枚舉構造。"
    }
  ],
  "symbols": [
    {
      "token": "I",
      "role": "first_isomorphism_operator",
      "role_zh": "第一同構定理構造",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "finite_group_homomorphism",
      "role_zh": "有限群同態",
      "scope": "representative_formula"
    },
    {
      "token": "h",
      "role": "induced_group_isomorphism",
      "role_zh": "誘導群同構",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-finite-group-homomorphism-kernel",
      "title_zh": "有限群同態的核",
      "reason_zh": "kernel與fiber cosets。"
    },
    {
      "id": "mko-finite-quotient-group",
      "title_zh": "有限商群",
      "reason_zh": "quotient carrier與well-defined multiplication。"
    },
    {
      "id": "mko-subgroup-homomorphism",
      "title_zh": "子群與群同態",
      "reason_zh": "image形成目標群子群。"
    },
    {
      "id": "mko-bijective",
      "title_zh": "雙射",
      "reason_zh": "induced map的injective與surjective。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "先資格核對homomorphism、kernel、image與quotient。",
      "代表元獨立性先於induced-map結論。",
      "完整驗同態與雙射，並保留image/codomain界線。"
    ],
    "common_misconception_zh": "只看cardinality相等就稱同構，或f未滿射時仍宣稱G/ker f≅Q。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪的完整有限枚舉證書不取代一般第一同構定理的抽象證明。",
    "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": "加入群中心與class equation等結構推論。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-first-isomorphism-theorem-finite-groups.md 整理；2026-10-15 脈衝 R01（使用者手動接續 2026-11-15 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"
  }
}