{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-finite-group-homomorphism-kernel",
  "semantic_family_id": "sf-finite-group-homomorphism-kernel",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限群同態的核"
  },
  "summary": {
    "zh-Hant": "有限群同態f:G→Q映到目標identity的完整原像是G的正規子群，且同態fibers恰為kernel的陪集。"
  },
  "statement": {
    "zh-Hant": "來源群與目標群都須由完整Cayley table認證，f須是complete total map並逐對滿足f(xy)=f(x)f(y)。kernel是目標identity的完整preimage，需直接驗subgroup與對所有g,k的gkg^{-1} closure。對任意a,b，f(a)=f(b)恰當a^{-1}b∈ker f，因此nonempty fibers是kernel cosets。kernel只含來源identity恰當f injective。非同態不得套kernel-normality；本節點不提前宣稱quotient或第一同構定理。"
  },
  "formula": {
    "tex": "K(f)=H",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>K</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": "K",
          "glyph": "K"
        },
        "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": "dd8ea093f6022b8ac0b2b3b7df25e1506a53aa1bc96a2a50229eb62c07378566"
    }
  },
  "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": "kernel、fibers、normality與injectivity均以完整有限枚舉核對。"
    }
  ],
  "symbols": [
    {
      "token": "K",
      "role": "kernel_operator",
      "role_zh": "核算子",
      "scope": "representative_formula"
    },
    {
      "token": "f",
      "role": "finite_group_homomorphism",
      "role_zh": "有限群同態",
      "scope": "representative_formula"
    },
    {
      "token": "H",
      "role": "normal_kernel_subgroup",
      "role_zh": "正規kernel子群",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-subgroup-homomorphism",
      "title_zh": "子群與群同態",
      "reason_zh": "homomorphism law與kernel背景。"
    },
    {
      "id": "mko-finite-normal-subgroup",
      "title_zh": "有限正規子群",
      "reason_zh": "kernel normality。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "complete total map。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "homomorphism qualification先於kernel結論。",
      "kernel normality以conjugation完整驗證。",
      "fiber-coset等價與trivial-kernel injectivity分開核對。"
    ],
    "common_misconception_zh": "對任意total map取identity preimage就稱正規kernel，或提前聲稱G/ker f同構於image。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本輪不重複一般kernel/image基礎題，專注normality、fibers與injectivity；第一同構定理留待後續。",
    "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-group-homomorphism-kernel.md 整理；2026-10-15 脈衝 R01（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"
  }
}