{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0080",
  "pilot_number": "R0050-25730",
  "number": 25730,
  "family_id": "qf-mod46-iso1-image-codomain-boundary",
  "title": {
    "zh-Hant": "第 25730 題：有限群第一同構定理",
    "en": "Question 25730: First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0080",
    "batch_id": "batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-first-isomorphism-theorem-finite-groups"
    ],
    "title_zh": "有限群第一同構定理",
    "stem_zh": "完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={\"source_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"target_identity\":0,\"mapping\":[0,2,4,6,8,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([0, 2, 4, 6, 8, 10], 12, 6, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)",
      "explanation_zh": "Wave6第九層核對=([0, 2, 4, 6, 8, 10], 12, 6, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)；依完整明示資料獨立重算；bar f(g ker f)=f(g)須對所有representatives well-defined並完整驗homomorphism與bijection-to-image；非滿射只能得到G/kernel同構於image；非同態不得套定理，有限枚舉不取代一般證明。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "有限群第一同構定理"
      ]
    },
    "variant": {
      "family_id": "qf-mod46-iso1-image-codomain-boundary",
      "seed": 202611151080,
      "parameters": {
        "source_table": [
          [
            0,
            1,
            2,
            3,
            4,
            5
          ],
          [
            1,
            2,
            3,
            4,
            5,
            0
          ],
          [
            2,
            3,
            4,
            5,
            0,
            1
          ],
          [
            3,
            4,
            5,
            0,
            1,
            2
          ],
          [
            4,
            5,
            0,
            1,
            2,
            3
          ],
          [
            5,
            0,
            1,
            2,
            3,
            4
          ]
        ],
        "source_identity": 0,
        "target_table": [
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7,
            8,
            9,
            10,
            11
          ],
          [
            1,
            2,
            3,
            4,
            5,
            6,
            7,
            8,
            9,
            10,
            11,
            0
          ],
          [
            2,
            3,
            4,
            5,
            6,
            7,
            8,
            9,
            10,
            11,
            0,
            1
          ],
          [
            3,
            4,
            5,
            6,
            7,
            8,
            9,
            10,
            11,
            0,
            1,
            2
          ],
          [
            4,
            5,
            6,
            7,
            8,
            9,
            10,
            11,
            0,
            1,
            2,
            3
          ],
          [
            5,
            6,
            7,
            8,
            9,
            10,
            11,
            0,
            1,
            2,
            3,
            4
          ],
          [
            6,
            7,
            8,
            9,
            10,
            11,
            0,
            1,
            2,
            3,
            4,
            5
          ],
          [
            7,
            8,
            9,
            10,
            11,
            0,
            1,
            2,
            3,
            4,
            5,
            6
          ],
          [
            8,
            9,
            10,
            11,
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          [
            9,
            10,
            11,
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7,
            8
          ],
          [
            10,
            11,
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7,
            8,
            9
          ],
          [
            11,
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7,
            8,
            9,
            10
          ]
        ],
        "target_identity": 0,
        "mapping": [
          0,
          2,
          4,
          6,
          8,
          10
        ],
        "domain": "Complete_finite_first_isomorphism",
        "operation": "iso1-image-codomain-boundary",
        "semantic_template": "qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]]}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_finite_or_symbolic_model",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "候選須主端重算；不構成人工審定、一般第一同構／strong duality／max-flow-min-cut證明、class equation、consistency、integrality、König theorem、無限domain computability、Lean或Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-15",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-0080",
    "batch_id": "batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "1b8c1ec9093762c448c89f3b4f66da4b06175af6e81dca9a8fd6485ec85f7ed1",
      "family_id": "qf-mod46-iso1-image-codomain-boundary",
      "seed": 202611151080,
      "semantic_template": "qf-mod46-iso1-image-codomain-boundary/{\"mapping\":[0,2,4,6,8,10],\"source_identity\":0,\"source_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"target_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]]}"
    },
    "fields": {
      "title": "First Isomorphism Reaches the Image, Not Automatically the Whole Codomain",
      "stem": "Use complete source Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary).",
      "answer": {
        "kind": "expression",
        "value": "([0, 2, 4, 6, 8, 10], 12, 6, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)",
        "explanation": "Compute the image and compare its size with the target group. The quotient is isomorphic to that image. Only when f is surjective does it identify the image with the codomain; finite-instance evidence does not replace the general abstract theorem proof. The exact result, retaining the requested order, is ([0, 2, 4, 6, 8, 10], 12, 6, No, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, No, finite_instance_certificate_not_a_general_theorem_proof)."
      },
      "skills": [
        "First Isomorphism Reaches the Image, Not Automatically the Whole Codomain"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "完整有限群、total homomorphism、kernel、image與quotient cosets的第一同構責任。操作=iso1-image-codomain-boundary；資料={\"source_table\":[[0,1,2,3,4,5],[1,2,3,4,5,0],[2,3,4,5,0,1],[3,4,5,0,1,2],[4,5,0,1,2,3],[5,0,1,2,3,4]],\"source_identity\":0,\"target_table\":[[0,1,2,3,4,5,6,7,8,9,10,11],[1,2,3,4,5,6,7,8,9,10,11,0],[2,3,4,5,6,7,8,9,10,11,0,1],[3,4,5,6,7,8,9,10,11,0,1,2],[4,5,6,7,8,9,10,11,0,1,2,3],[5,6,7,8,9,10,11,0,1,2,3,4],[6,7,8,9,10,11,0,1,2,3,4,5],[7,8,9,10,11,0,1,2,3,4,5,6],[8,9,10,11,0,1,2,3,4,5,6,7],[9,10,11,0,1,2,3,4,5,6,7,8],[10,11,0,1,2,3,4,5,6,7,8,9],[11,0,1,2,3,4,5,6,7,8,9,10]],\"target_identity\":0,\"mapping\":[0,2,4,6,8,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use complete source Cayley table=[[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], identity=0, target table=[[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], identity=0, and the aligned total mapping=[0, 2, 4, 6, 8, 10]. Table row x/column y is x*y. Qualify both groups, then every homomorphism product pair before invoking the theorem. Give (image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [iso1-image-codomain-boundary]",
      "after": "derived = [[image, codomain_order, image_order, surjective_to_codomain, image_isomorphism_statement, codomain_scope, general_theorem_proof_claim, boundary]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "直接列完整來源欄序，不需JavaScript。LP weak為13欄、zero與strong為15欄，前提與角色依原operation／mode，不補欄。",
        "en": "List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [Iso1 ORIGINAL SOURCE, [[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], 0]",
      "after": "derived = [6, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5]], 是]",
      "relation": "implies",
      "rule_id": "Group_complete_table_actual_identity_bound64",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整closed Cayley table與實際identity先資格核對，群階可到64，不假設identity0。",
        "en": "Qualify the complete closed Cayley table and actual identity first,allowing order64 without assuming identity0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [Iso1 ORIGINAL SOURCE, 6, 0]",
      "after": "derived = [[[是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是]], 216, [0, 5, 4, 3, 2, 1]]",
      "relation": "implies",
      "rule_id": "Group_ALL_triples_compact_pair_matrix_and_ALL_inverses",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每pair格逐全部第三元素核結合律，完整n³而非抽樣；inverse同時左右回實際identity。",
        "en": "Each pair-matrix entry checks every third element for associativity,covering all n³ triples,not samples.Inverses return the actual identity from both sides."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [Iso1 ORIGINAL TARGET, [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1], [3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2], [4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3], [5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4], [6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5], [7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6], [8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7], [9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8], [10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]], 0]",
      "after": "derived = [12, [[0, 0, 0, 0], [1, 1, 1, 1], [2, 2, 2, 2], [3, 3, 3, 3], [4, 4, 4, 4], [5, 5, 5, 5], [6, 6, 6, 6], [7, 7, 7, 7], [8, 8, 8, 8], [9, 9, 9, 9], [10, 10, 10, 10], [11, 11, 11, 11]], 是]",
      "relation": "implies",
      "rule_id": "Group_complete_table_actual_identity_bound64",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整closed Cayley table與實際identity先資格核對，群階可到64，不假設identity0。",
        "en": "Qualify the complete closed Cayley table and actual identity first,allowing order64 without assuming identity0."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [Iso1 ORIGINAL TARGET, 12, 0]",
      "after": "derived = [[[是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是], [是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是, 是]], 1728, [0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]]",
      "relation": "implies",
      "rule_id": "Group_ALL_triples_compact_pair_matrix_and_ALL_inverses",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每pair格逐全部第三元素核結合律，完整n³而非抽樣；inverse同時左右回實際identity。",
        "en": "Each pair-matrix entry checks every third element for associativity,covering all n³ triples,not samples.Inverses return the actual identity from both sides."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [Iso1 ACTUAL kernel, [0], 0]",
      "after": "derived = [是, [[0, 0, 0, 是]], 是, [[0, 0, 是]], 是, 是, [[0, [0], 是], [1, [0], 是], [2, [0], 是], [3, [0], 是], [4, [0], 是], [5, [0], 是]], 是]",
      "relation": "implies",
      "rule_id": "Subgroup_actual_identity_product_inverse_ALL_conjugates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "含identity、全部product與inverse closure各核，三者合取才成subgroup；再逐全部g驗normal。即使某map失敗，這些實際property仍可分別True，不由定理名稱填旗標。",
        "en": "Check identity,all products and inverse closure separately before subgroup qualification,then every g for normality.These are actual properties that may individually hold even when a map fails;do not fill flags from a theorem name."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "source roles = [Iso1 ACTUAL image, [0, 2, 4, 6, 8, 10], 0]",
      "after": "derived = [是, [[0, 0, 0, 是], [0, 2, 2, 是], [0, 4, 4, 是], [0, 6, 6, 是], [0, 8, 8, 是], [0, 10, 10, 是], [2, 0, 2, 是], [2, 2, 4, 是], [2, 4, 6, 是], [2, 6, 8, 是], [2, 8, 10, 是], [2, 10, 0, 是], [4, 0, 4, 是], [4, 2, 6, 是], [4, 4, 8, 是], [4, 6, 10, 是], [4, 8, 0, 是], [4, 10, 2, 是], [6, 0, 6, 是], [6, 2, 8, 是], [6, 4, 10, 是], [6, 6, 0, 是], [6, 8, 2, 是], [6, 10, 4, 是], [8, 0, 8, 是], [8, 2, 10, 是], [8, 4, 0, 是], [8, 6, 2, 是], [8, 8, 4, 是], [8, 10, 6, 是], [10, 0, 10, 是], [10, 2, 0, 是], [10, 4, 2, 是], [10, 6, 4, 是], [10, 8, 6, 是], [10, 10, 8, 是]], 是, [[0, 0, 是], [2, 10, 是], [4, 8, 是], [6, 6, 是], [8, 4, 是], [10, 2, 是]], 是, 是, [[0, [0, 2, 4, 6, 8, 10], 是], [1, [0, 2, 4, 6, 8, 10], 是], [2, [0, 2, 4, 6, 8, 10], 是], [3, [0, 2, 4, 6, 8, 10], 是], [4, [0, 2, 4, 6, 8, 10], 是], [5, [0, 2, 4, 6, 8, 10], 是], [6, [0, 2, 4, 6, 8, 10], 是], [7, [0, 2, 4, 6, 8, 10], 是], [8, [0, 2, 4, 6, 8, 10], 是], [9, [0, 2, 4, 6, 8, 10], 是], [10, [0, 2, 4, 6, 8, 10], 是], [11, [0, 2, 4, 6, 8, 10], 是]], 是]",
      "relation": "implies",
      "rule_id": "Subgroup_actual_identity_product_inverse_ALL_conjugates",
      "conditions": [],
      "explanation": {
        "zh-Hant": "含identity、全部product與inverse closure各核，三者合取才成subgroup；再逐全部g驗normal。即使某map失敗，這些實際property仍可分別True，不由定理名稱填旗標。",
        "en": "Check identity,all products and inverse closure separately before subgroup qualification,then every g for normality.These are actual properties that may individually hold even when a map fails;do not fill flags from a theorem name."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "source roles = [[0, 2, 4, 6, 8, 10], 0, 0]",
      "after": "derived = [[[0, 0, 0, 0, 是], [0, 1, 2, 2, 是], [0, 2, 4, 4, 是], [0, 3, 6, 6, 是], [0, 4, 8, 8, 是], [0, 5, 10, 10, 是], [1, 0, 2, 2, 是], [1, 1, 4, 4, 是], [1, 2, 6, 6, 是], [1, 3, 8, 8, 是], [1, 4, 10, 10, 是], [1, 5, 0, 0, 是], [2, 0, 4, 4, 是], [2, 1, 6, 6, 是], [2, 2, 8, 8, 是], [2, 3, 10, 10, 是], [2, 4, 0, 0, 是], [2, 5, 2, 2, 是], [3, 0, 6, 6, 是], [3, 1, 8, 8, 是], [3, 2, 10, 10, 是], [3, 3, 0, 0, 是], [3, 4, 2, 2, 是], [3, 5, 4, 4, 是], [4, 0, 8, 8, 是], [4, 1, 10, 10, 是], [4, 2, 0, 0, 是], [4, 3, 2, 2, 是], [4, 4, 4, 4, 是], [4, 5, 6, 6, 是], [5, 0, 10, 10, 是], [5, 1, 0, 0, 是], [5, 2, 2, 2, 是], [5, 3, 4, 4, 是], [5, 4, 6, 6, 是], [5, 5, 8, 8, 是]], 是, null, [0], [0, 2, 4, 6, 8, 10], 是, 是]",
      "relation": "implies",
      "rule_id": "Iso1_ALL_product_laws_ACTUAL_kernel_image_properties_before_theorem",
      "conditions": [],
      "explanation": {
        "zh-Hant": "全pair驗同態，並取完整target identity preimage與image。qualification允許bad map且property按實際核，可能同態False而kernel normal／image subgroup True；空kernel不補identity fallback。",
        "en": "Audit every homomorphism pair and the full target-identity preimage/image.Qualification allows a bad map and reports actual properties:homFalse can coexist with normal kernel or subgroup image.An empty kernel is not replaced by an identity fallback."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "source roles = [[0], 是, [0, 2, 4, 6, 8, 10]]",
      "after": "derived = [[[0], [1], [2], [3], [4], [5]], [0, 1, 2, 3, 4, 5], [[0, 1, 2, 3, 4, 5], [1, 2, 3, 4, 5, 0], [2, 3, 4, 5, 0, 1], [3, 4, 5, 0, 1, 2], [4, 5, 0, 1, 2, 3], [5, 0, 1, 2, 3, 4]], [[0, 0, [0], 是], [0, 1, [1], 是], [0, 2, [2], 是], [0, 3, [3], 是], [0, 4, [4], 是], [0, 5, [5], 是], [1, 0, [1], 是], [1, 1, [2], 是], [1, 2, [3], 是], [1, 3, [4], 是], [1, 4, [5], 是], [1, 5, [0], 是], [2, 0, [2], 是], [2, 1, [3], 是], [2, 2, [4], 是], [2, 3, [5], 是], [2, 4, [0], 是], [2, 5, [1], 是], [3, 0, [3], 是], [3, 1, [4], 是], [3, 2, [5], 是], [3, 3, [0], 是], [3, 4, [1], 是], [3, 5, [2], 是], [4, 0, [4], 是], [4, 1, [5], 是], [4, 2, [0], 是], [4, 3, [1], 是], [4, 4, [2], 是], [4, 5, [3], 是], [5, 0, [5], 是], [5, 1, [0], 是], [5, 2, [1], 是], [5, 3, [2], 是], [5, 4, [3], 是], [5, 5, [4], 是]], [[0, [0], [0], 是], [1, [1], [2], 是], [2, [2], [4], 是], [3, [3], [6], 是], [4, [4], [8], 是], [5, [5], [10], 是]], 是]",
      "relation": "implies",
      "rule_id": "Iso1_full_LEFT_coset_quotient_ALL_products_and_representative_images",
      "conditions": [],
      "explanation": {
        "zh-Hant": "kernel normal使(ah)(bk)=ab(b^-1hb)k∈abK，逐全部代表元驗product coset相同；同一coset的每個f(g)也須同值才定義bar_f。carrier是coset SETS而非首代表元。",
        "en": "Kernel normality gives(ah)(bk)=ab(b^-1hb)k in abK;check every representative product agrees on its quotient cell.Every f(g) in a coset must also agree before defining bar_f.The carrier consists of coset SETS,not least representatives."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S010",
      "uses": [
        "G001",
        "S009"
      ],
      "before": "source roles = [[[0], [1], [2], [3], [4], [5]], [0, 2, 4, 6, 8, 10], [0, 2, 4, 6, 8, 10]]",
      "after": "derived = [[[0, 0, 0, 0, 是], [0, 1, 2, 2, 是], [0, 2, 4, 4, 是], [0, 3, 6, 6, 是], [0, 4, 8, 8, 是], [0, 5, 10, 10, 是], [1, 0, 2, 2, 是], [1, 1, 4, 4, 是], [1, 2, 6, 6, 是], [1, 3, 8, 8, 是], [1, 4, 10, 10, 是], [1, 5, 0, 0, 是], [2, 0, 4, 4, 是], [2, 1, 6, 6, 是], [2, 2, 8, 8, 是], [2, 3, 10, 10, 是], [2, 4, 0, 0, 是], [2, 5, 2, 2, 是], [3, 0, 6, 6, 是], [3, 1, 8, 8, 是], [3, 2, 10, 10, 是], [3, 3, 0, 0, 是], [3, 4, 2, 2, 是], [3, 5, 4, 4, 是], [4, 0, 8, 8, 是], [4, 1, 10, 10, 是], [4, 2, 0, 0, 是], [4, 3, 2, 2, 是], [4, 4, 4, 4, 是], [4, 5, 6, 6, 是], [5, 0, 10, 10, 是], [5, 1, 0, 0, 是], [5, 2, 2, 2, 是], [5, 3, 4, 4, 是], [5, 4, 6, 6, 是], [5, 5, 8, 8, 是]], 是, 是, 是, 是, 否]",
      "relation": "implies",
      "rule_id": "Iso1_ALL_induced_product_pairs_injective_and_onto_IMAGE_separate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "bar_f保乘法由f(ab)=f(a)f(b)；若f(a)=f(b)，a^-1b∈K故同coset，給injective。任意image值f(g)由gK命中，給onto IMAGE；不由同cardinality代替這些責任，也不自動onto整個target。",
        "en": "The identity f(ab)=f(a)f(b) preserves products.If f(a)=f(b),a^-1b is in K and the cosets agree,giving injectivity.Every image value f(g) is hit by gK,giving surjectivity onto the IMAGE.Equal cardinalities do not replace these obligations or give surjectivity to the whole target."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S011",
      "uses": [
        "G001",
        "S010"
      ],
      "before": "source roles = [[0, 2, 4, 6, 8, 10], 12, 6]",
      "after": "derived = [否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否]",
      "relation": "implies",
      "rule_id": "Iso1_image_VS_codomain_scope_and_finite_general_proof_boundary",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只有實際f滿射，image才等同整個codomain。最後No保一般抽象證明／形式化責任，不否定本題完整有限同構證書或教學理由。",
        "en": "Only actual surjectivity identifies image with the whole codomain.The final No preserves general abstract/formal proof obligations,not denial of this complete finite isomorphism certificate or its pedagogical argument."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        0,
        2,
        4,
        6,
        8,
        10
      ],
      12,
      6,
      false,
      "G/kernel_is_isomorphic_to_image",
      "non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain",
      false,
      "finite_instance_certificate_not_a_general_theorem_proof"
    ]
  },
  "conclusion": {
    "canonical_value": "([0, 2, 4, 6, 8, 10], 12, 6, 否, G/kernel_is_isomorphic_to_image, non_surjective_f_does_not_identify_the_quotient_with_the_whole_codomain, 否, finite_instance_certificate_not_a_general_theorem_proof)",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations/questions.jsonl",
      "batch_sha256": "b27c8fc97af12e9d77e259f367fbae66d666796519a9c4014e73f875c727b827",
      "original_line": 80,
      "original_line_sha256": "50c71ae6446c36e65bccbddc699c3e4d6e68a0c04b237129bbc4a90855194ee6",
      "question_compact_sha256": "1b8c1ec9093762c448c89f3b4f66da4b06175af6e81dca9a8fd6485ec85f7ed1",
      "english_row_compact_sha256": "5f4f93a867c9cf3a1edb54a3be208d2e43d9c1963c151de7381b387b48c603b5",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261115-r01-first-isomorphism-theorem-finite-groups-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-first-isomorphism-theorem-finite-groups": {
        "sha256": "5586ce5ed3d47467ffd6c485dfcb495844143a17783d770ce2572420bfa28065",
        "english_sha256": "f9e622d65b015b588d1e29eb89655426b79abaf193e2f4763bec7b9e8c6aeadb",
        "original_path": "public/data/mko/first-isomorphism-theorem-finite-groups.json",
        "english_path": "public/data/i18n/editions/mko-first-isomorphism-theorem-finite-groups.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "4624c29c8cf22048332daea19f430f922e032c30410aa690ba3f1924d8b9ee45",
    "original_candidate_sha256": "61f388141df20512716ea64b712a4bf959a87f907ddab50bf123b3f99e2c6870",
    "original_candidate_line_sha256": "61f388141df20512716ea64b712a4bf959a87f907ddab50bf123b3f99e2c6870"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-010-p51",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82",
      "question_first_witness_sha256": "6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 11,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "28725fb3ade638836a3848b0bf01aeba3502e3af9784893e54ae3eda3d3df5bd",
      "arithmetic_steps": 0
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}