{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-subgroup-homomorphism",
  "semantic_family_id": "sf-subgroup-homomorphism",
  "type": "concept",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "子群與群同態"
  },
  "summary": {
    "zh-Hant": "子群在繼承運算下自身成群；群同態保持乘法，kernel與image由完整映射表導出。"
  },
  "statement": {
    "zh-Hant": "有限群G的非空子集H若對運算封閉、含單位元且對逆元封閉，則為子群。群同態f:G→K滿足f(a*b)=f(a)*f(b)；同態不自動為單射。"
  },
  "formula": {
    "tex": "H(G)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>H</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "H",
        "glyph": "H"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "G",
          "glyph": "G"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "dc00d660515759496f17320c50531cfbde3678cb84986919bbe1350cdbfa8fe4"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "題庫提供來源與目標的完整有限Cayley表及完整映射表。"
    },
    {
      "id": "a2",
      "zh-Hant": "結論只屬指定有限群；不外推任意群，也不把同態自動當同構。"
    }
  ],
  "symbols": [
    {
      "token": "H",
      "role": "subgroup_or_homomorphism_record",
      "role_zh": "子群／同態紀錄",
      "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": "子群與同態以已驗群結構為前提。"
    },
    {
      "id": "mko-subset",
      "title_zh": "子集合",
      "reason_zh": "子群是群載體的子集合。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "同態是保持運算的總映射。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "有限子群可核閉合、單位元與逆元。",
      "同態責任需逐一核對全部輸入對。",
      "kernel是映到目標單位元的來源元素；image是所有映射值，image大小等於來源大小才可能單射。"
    ],
    "common_misconception_zh": "只核閉合就叫子群，或把每個同態都當單射／同構。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "完整有限表只證指定模型。",
    "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": "加入陪集、正規子群、商群與同構定理。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-subgroup-homomorphism.md 整理；2026-10-13 脈衝 R03（使用者手動接續 2026-10-23 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"
  }
}