{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261015-advance-20261113-r01-finite-normal-subgroup-0068",
  "pilot_number": "R0048-24543",
  "number": 24543,
  "family_id": "qf-mod44-normal-left-right-coset-audit",
  "title": {
    "zh-Hant": "第 24543 題：有限正規子群",
    "en": "Question 24543: Left and Right Cosets Must Agree for Every g"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261015-advance-20261113-r01-finite-normal-subgroup-0068",
    "batch_id": "batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-finite-normal-subgroup"
    ],
    "title_zh": "有限正規子群",
    "stem_zh": "完整Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={\"group_table\":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3]],\"identity\":3,\"subset\":[0,1,2,3,4,5,6,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
    "answer": {
      "kind": "expression",
      "value": "([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)",
      "explanation_zh": "Wave6第七層核對=([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)；依完整明示資料獨立重算；normal必須完整驗gHg^-1=H或逐g左／右陪集相等；trivial與whole subgroup保留；抽樣共軛、有限性或order divisibility不足；本節點尚不宣稱quotient multiplication well-defined。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 3,
      "skills": [
        "有限正規子群"
      ]
    },
    "variant": {
      "family_id": "qf-mod44-normal-left-right-coset-audit",
      "seed": 202611131068,
      "parameters": {
        "group_table": [
          [
            5,
            6,
            7,
            0,
            1,
            2,
            3,
            4
          ],
          [
            6,
            7,
            0,
            1,
            2,
            3,
            4,
            5
          ],
          [
            7,
            0,
            1,
            2,
            3,
            4,
            5,
            6
          ],
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          [
            1,
            2,
            3,
            4,
            5,
            6,
            7,
            0
          ],
          [
            2,
            3,
            4,
            5,
            6,
            7,
            0,
            1
          ],
          [
            3,
            4,
            5,
            6,
            7,
            0,
            1,
            2
          ],
          [
            4,
            5,
            6,
            7,
            0,
            1,
            2,
            3
          ]
        ],
        "identity": 3,
        "subset": [
          0,
          1,
          2,
          3,
          4,
          5,
          6,
          7
        ],
        "domain": "Complete_finite_group_normality",
        "operation": "normal-left-right-coset-audit",
        "semantic_template": "qf-mod44-normal-left-right-coset-audit/{\"group_table\":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3]],\"identity\":3,\"subset\":[0,1,2,3,4,5,6,7]}"
      }
    },
    "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、sampled normality、finite-sum exponential、formal-only analytic equality、circular Euler criterion、finite-trace decider、stationary extremum、feasible maximum-flow、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-20261113-r01-finite-normal-subgroup-0068",
    "batch_id": "batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "e7d578be62eb6fabd111abedea97d7b806d6c04ed77ea040db210d94ba89f115",
      "family_id": "qf-mod44-normal-left-right-coset-audit",
      "seed": 202611131068,
      "semantic_template": "qf-mod44-normal-left-right-coset-audit/{\"group_table\":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3]],\"identity\":3,\"subset\":[0,1,2,3,4,5,6,7]}"
    },
    "fields": {
      "title": "Left and Right Cosets Must Agree for Every g",
      "stem": "Use complete finite Cayley table=[[5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3]], identity=3 and candidate subset H=[0, 1, 2, 3, 4, 5, 6, 7]. Row g/column h gives g*h. Validate all group axioms, then identity/product/inverse closure of the whole subset before treating it as a subgroup. Normality has a separate all-g obligation. Give (subgroup, coset_rows, all_left_right_equal, certificate).",
      "answer": {
        "kind": "expression",
        "value": "([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], Yes], [1, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], Yes], [2, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], Yes], [3, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], Yes], [4, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], Yes], [5, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], Yes], [6, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], Yes], [7, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], Yes]], Yes, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)",
        "explanation": "Require subgroup qualification and compare the entire gH and Hg for each g in group-index order. Their all-g equality is equivalent to normality. Do not infer well-defined quotient multiplication here or conflate nonabelian left/right cosets. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], Yes], [1, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], Yes], [2, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], Yes], [3, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], Yes], [4, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], Yes], [5, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], Yes], [6, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], Yes], [7, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], Yes]], Yes, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)."
      },
      "skills": [
        "Left and Right Cosets Must Agree for Every g"
      ],
      "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Cayley table、完整subgroup資格與forall g的normality責任。操作=normal-left-right-coset-audit；資料={\"group_table\":[[5,6,7,0,1,2,3,4],[6,7,0,1,2,3,4,5],[7,0,1,2,3,4,5,6],[0,1,2,3,4,5,6,7],[1,2,3,4,5,6,7,0],[2,3,4,5,6,7,0,1],[3,4,5,6,7,0,1,2],[4,5,6,7,0,1,2,3]],\"identity\":3,\"subset\":[0,1,2,3,4,5,6,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。",
        "en": "Use complete finite Cayley table=[[5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3]], identity=3 and candidate subset H=[0, 1, 2, 3, 4, 5, 6, 7]. Row g/column h gives g*h. Validate all group axioms, then identity/product/inverse closure of the whole subset before treating it as a subgroup. Normality has a separate all-g obligation. Give (subgroup, coset_rows, all_left_right_equal, certificate)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [normal-left-right-coset-audit]",
      "after": "derived = [[subgroup, coset_rows, all_left_right_equal, certificate]]",
      "relation": "implies",
      "rule_id": "source_ordered_fields_without_JS",
      "conditions": [],
      "explanation": {
        "zh-Hant": "直接列來源完整欄序，不需JavaScript；原題、布林旗標、數學物件與非聲稱保持分層。",
        "en": "List the full source field order without JavaScript,keeping original data,Boolean flags,mathematical objects and nonclaims distinct."
      },
      "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 = [Normality ORIGINAL group, 32, [[5, 6, 7, 0, 1, 2, 3, 4], [6, 7, 0, 1, 2, 3, 4, 5], [7, 0, 1, 2, 3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6, 7], [1, 2, 3, 4, 5, 6, 7, 0], [2, 3, 4, 5, 6, 7, 0, 1], [3, 4, 5, 6, 7, 0, 1, 2], [4, 5, 6, 7, 0, 1, 2, 3]], 3]",
      "after": "derived = [8, [[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]], 是]",
      "relation": "implies",
      "rule_id": "Group_complete_Cayley_signature_actual_identity",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先驗完整n×n閉合表與實際identity標籤，非預設0。mod44上限32，mod45上限64；同一carrier不能混入另一表。",
        "en": "Validate the complete closed n-by-n table and actual identity label,not an assumed0.Mod44 allows32 and mod45 allows64;do not mix carriers or tables."
      },
      "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 = [Normality ORIGINAL group, 8, FOR EVERY x,y,z in0..n-1]",
      "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]], 512, 是]",
      "relation": "implies",
      "rule_id": "Group_ALL_associativity_triples_compact_complete_pair_matrix",
      "conditions": [],
      "explanation": {
        "zh-Hant": "矩陣(x,y)逐全部z核(xy)z=x(yz)，故涵蓋全部n³三元組，不是抽樣。表已在前步完整列出，不重印數百萬同義查表項。",
        "en": "Entry(x,y) checks (xy)z=x(yz) for every z,covering all n^3 triples,not samples.The complete table is already displayed,so do not print millions of redundant lookups."
      },
      "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 = [Normality ORIGINAL group, 3]",
      "after": "derived = [[6, 5, 4, 3, 2, 1, 0, 7], [[0, 6, 3, 3], [1, 5, 3, 3], [2, 4, 3, 3], [3, 3, 3, 3], [4, 2, 3, 3], [5, 1, 3, 3], [6, 0, 3, 3], [7, 7, 3, 3]], 是]",
      "relation": "implies",
      "rule_id": "Group_ALL_two_sided_inverses",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個元素的左、右逆都要回到同一實際identity；結合律使兩側逆唯一。這才完成群前提。",
        "en": "Every left and right inverse must return the same actual identity;associativity gives uniqueness.This completes the group premises."
      },
      "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 = [Normality ORIGINAL H, [0, 1, 2, 3, 4, 5, 6, 7], 3]",
      "after": "derived = [是, [[0, 0, 5, 是], [0, 1, 6, 是], [0, 2, 7, 是], [0, 3, 0, 是], [0, 4, 1, 是], [0, 5, 2, 是], [0, 6, 3, 是], [0, 7, 4, 是], [1, 0, 6, 是], [1, 1, 7, 是], [1, 2, 0, 是], [1, 3, 1, 是], [1, 4, 2, 是], [1, 5, 3, 是], [1, 6, 4, 是], [1, 7, 5, 是], [2, 0, 7, 是], [2, 1, 0, 是], [2, 2, 1, 是], [2, 3, 2, 是], [2, 4, 3, 是], [2, 5, 4, 是], [2, 6, 5, 是], [2, 7, 6, 是], [3, 0, 0, 是], [3, 1, 1, 是], [3, 2, 2, 是], [3, 3, 3, 是], [3, 4, 4, 是], [3, 5, 5, 是], [3, 6, 6, 是], [3, 7, 7, 是], [4, 0, 1, 是], [4, 1, 2, 是], [4, 2, 3, 是], [4, 3, 4, 是], [4, 4, 5, 是], [4, 5, 6, 是], [4, 6, 7, 是], [4, 7, 0, 是], [5, 0, 2, 是], [5, 1, 3, 是], [5, 2, 4, 是], [5, 3, 5, 是], [5, 4, 6, 是], [5, 5, 7, 是], [5, 6, 0, 是], [5, 7, 1, 是], [6, 0, 3, 是], [6, 1, 4, 是], [6, 2, 5, 是], [6, 3, 6, 是], [6, 4, 7, 是], [6, 5, 0, 是], [6, 6, 1, 是], [6, 7, 2, 是], [7, 0, 4, 是], [7, 1, 5, 是], [7, 2, 6, 是], [7, 3, 7, 是], [7, 4, 0, 是], [7, 5, 1, 是], [7, 6, 2, 是], [7, 7, 3, 是]], 是, [[0, 6, 是], [1, 5, 是], [2, 4, 是], [3, 3, 是], [4, 2, 是], [5, 1, 是], [6, 0, 是], [7, 7, 是]], 是, 是]",
      "relation": "implies",
      "rule_id": "Subgroup_identity_ALL_products_ALL_inverses_separate",
      "conditions": [],
      "explanation": {
        "zh-Hant": "H非空且無重複；含identity、任兩元素乘積、每個逆元分別核全域，三者合取才可稱subgroup。order divisibility或有限性不足。",
        "en": "H is nonempty and unique.Check identity membership,all products and every inverse separately;their conjunction qualifies a subgroup.Size divisibility or finiteness alone is insufficient."
      },
      "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 = [Normality ORIGINAL H, 0, [0, 1, 2, 3, 4, 5, 6, 7], 6]",
      "after": "derived = [[[0, 5, 6, 0], [1, 6, 6, 1], [2, 7, 6, 2], [3, 0, 6, 3], [4, 1, 6, 4], [5, 2, 6, 5], [6, 3, 6, 6], [7, 4, 6, 7]], [0, 1, 2, 3, 4, 5, 6, 7], 是]",
      "relation": "implies",
      "rule_id": "Subgroup_EVERY_g_entire_conjugate_set",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。",
        "en": "In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators."
      },
      "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 = [Normality ORIGINAL H, 1, [0, 1, 2, 3, 4, 5, 6, 7], 5]",
      "after": "derived = [[[0, 6, 5, 0], [1, 7, 5, 1], [2, 0, 5, 2], [3, 1, 5, 3], [4, 2, 5, 4], [5, 3, 5, 5], [6, 4, 5, 6], [7, 5, 5, 7]], [0, 1, 2, 3, 4, 5, 6, 7], 是]",
      "relation": "implies",
      "rule_id": "Subgroup_EVERY_g_entire_conjugate_set",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。",
        "en": "In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators."
      },
      "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 = [Normality ORIGINAL H, 2, [0, 1, 2, 3, 4, 5, 6, 7], 4]",
      "after": "derived = [[[0, 7, 4, 0], [1, 0, 4, 1], [2, 1, 4, 2], [3, 2, 4, 3], [4, 3, 4, 4], [5, 4, 4, 5], [6, 5, 4, 6], [7, 6, 4, 7]], [0, 1, 2, 3, 4, 5, 6, 7], 是]",
      "relation": "implies",
      "rule_id": "Subgroup_EVERY_g_entire_conjugate_set",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。",
        "en": "In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators."
      },
      "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 = [Normality ORIGINAL H, 3, [0, 1, 2, 3, 4, 5, 6, 7], 3]",
      "after": "derived = [[[0, 0, 3, 0], [1, 1, 3, 1], [2, 2, 3, 2], [3, 3, 3, 3], [4, 4, 3, 4], [5, 5, 3, 5], [6, 6, 3, 6], [7, 7, 3, 7]], [0, 1, 2, 3, 4, 5, 6, 7], 是]",
      "relation": "implies",
      "rule_id": "Subgroup_EVERY_g_entire_conjugate_set",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。",
        "en": "In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators."
      },
      "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 = [Normality ORIGINAL H, 4, [0, 1, 2, 3, 4, 5, 6, 7], 2]",
      "after": "derived = [[[0, 1, 2, 0], [1, 2, 2, 1], [2, 3, 2, 2], [3, 4, 2, 3], [4, 5, 2, 4], [5, 6, 2, 5], [6, 7, 2, 6], [7, 0, 2, 7]], [0, 1, 2, 3, 4, 5, 6, 7], 是]",
      "relation": "implies",
      "rule_id": "Subgroup_EVERY_g_entire_conjugate_set",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。",
        "en": "In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators."
      },
      "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 = [Normality ORIGINAL H, 5, [0, 1, 2, 3, 4, 5, 6, 7], 1]",
      "after": "derived = [[[0, 2, 1, 0], [1, 3, 1, 1], [2, 4, 1, 2], [3, 5, 1, 3], [4, 6, 1, 4], [5, 7, 1, 5], [6, 0, 1, 6], [7, 1, 1, 7]], [0, 1, 2, 3, 4, 5, 6, 7], 是]",
      "relation": "implies",
      "rule_id": "Subgroup_EVERY_g_entire_conjugate_set",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。",
        "en": "In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators."
      },
      "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": "S012",
      "uses": [
        "G001",
        "S011"
      ],
      "before": "source roles = [Normality ORIGINAL H, 6, [0, 1, 2, 3, 4, 5, 6, 7], 0]",
      "after": "derived = [[[0, 3, 0, 0], [1, 4, 0, 1], [2, 5, 0, 2], [3, 6, 0, 3], [4, 7, 0, 4], [5, 0, 0, 5], [6, 1, 0, 6], [7, 2, 0, 7]], [0, 1, 2, 3, 4, 5, 6, 7], 是]",
      "relation": "implies",
      "rule_id": "Subgroup_EVERY_g_entire_conjugate_set",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。",
        "en": "In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators."
      },
      "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": "S013",
      "uses": [
        "G001",
        "S012"
      ],
      "before": "source roles = [Normality ORIGINAL H, 7, [0, 1, 2, 3, 4, 5, 6, 7], 7]",
      "after": "derived = [[[0, 4, 7, 0], [1, 5, 7, 1], [2, 6, 7, 2], [3, 7, 7, 3], [4, 0, 7, 4], [5, 1, 7, 5], [6, 2, 7, 6], [7, 3, 7, 7]], [0, 1, 2, 3, 4, 5, 6, 7], 是]",
      "relation": "implies",
      "rule_id": "Subgroup_EVERY_g_entire_conjugate_set",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按同一表先乘g*h，再乘g的右逆，遍歷全部h；再對全部g核集合等式，不由取樣或生成元猜normal。",
        "en": "In the same table compute g*h then multiply by g inverse for every h.Check set equality for every g,not sampled elements or unproved generators."
      },
      "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": "S014",
      "uses": [
        "G001",
        "S013"
      ],
      "before": "source roles = [0, [0, 1, 2, 3, 4, 5, 6, 7]]",
      "after": "derived = [[0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]",
      "relation": "implies",
      "rule_id": "Normality_ALL_g_LEFT_RIGHT_cosets",
      "conditions": [],
      "explanation": {
        "zh-Hant": "gH與Hg是同一g的完整左／右陪集；非交換群不能偷換乘法方向。",
        "en": "These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group."
      },
      "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": "S015",
      "uses": [
        "G001",
        "S014"
      ],
      "before": "source roles = [1, [0, 1, 2, 3, 4, 5, 6, 7]]",
      "after": "derived = [[0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]",
      "relation": "implies",
      "rule_id": "Normality_ALL_g_LEFT_RIGHT_cosets",
      "conditions": [],
      "explanation": {
        "zh-Hant": "gH與Hg是同一g的完整左／右陪集；非交換群不能偷換乘法方向。",
        "en": "These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group."
      },
      "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": "S016",
      "uses": [
        "G001",
        "S015"
      ],
      "before": "source roles = [2, [0, 1, 2, 3, 4, 5, 6, 7]]",
      "after": "derived = [[0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]",
      "relation": "implies",
      "rule_id": "Normality_ALL_g_LEFT_RIGHT_cosets",
      "conditions": [],
      "explanation": {
        "zh-Hant": "gH與Hg是同一g的完整左／右陪集；非交換群不能偷換乘法方向。",
        "en": "These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group."
      },
      "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": "S017",
      "uses": [
        "G001",
        "S016"
      ],
      "before": "source roles = [3, [0, 1, 2, 3, 4, 5, 6, 7]]",
      "after": "derived = [[0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]",
      "relation": "implies",
      "rule_id": "Normality_ALL_g_LEFT_RIGHT_cosets",
      "conditions": [],
      "explanation": {
        "zh-Hant": "gH與Hg是同一g的完整左／右陪集；非交換群不能偷換乘法方向。",
        "en": "These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group."
      },
      "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": "S018",
      "uses": [
        "G001",
        "S017"
      ],
      "before": "source roles = [4, [0, 1, 2, 3, 4, 5, 6, 7]]",
      "after": "derived = [[0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]",
      "relation": "implies",
      "rule_id": "Normality_ALL_g_LEFT_RIGHT_cosets",
      "conditions": [],
      "explanation": {
        "zh-Hant": "gH與Hg是同一g的完整左／右陪集；非交換群不能偷換乘法方向。",
        "en": "These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group."
      },
      "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": "S019",
      "uses": [
        "G001",
        "S018"
      ],
      "before": "source roles = [5, [0, 1, 2, 3, 4, 5, 6, 7]]",
      "after": "derived = [[0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]",
      "relation": "implies",
      "rule_id": "Normality_ALL_g_LEFT_RIGHT_cosets",
      "conditions": [],
      "explanation": {
        "zh-Hant": "gH與Hg是同一g的完整左／右陪集；非交換群不能偷換乘法方向。",
        "en": "These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group."
      },
      "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": "S020",
      "uses": [
        "G001",
        "S019"
      ],
      "before": "source roles = [6, [0, 1, 2, 3, 4, 5, 6, 7]]",
      "after": "derived = [[0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]",
      "relation": "implies",
      "rule_id": "Normality_ALL_g_LEFT_RIGHT_cosets",
      "conditions": [],
      "explanation": {
        "zh-Hant": "gH與Hg是同一g的完整左／右陪集；非交換群不能偷換乘法方向。",
        "en": "These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group."
      },
      "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": "S021",
      "uses": [
        "G001",
        "S020"
      ],
      "before": "source roles = [7, [0, 1, 2, 3, 4, 5, 6, 7]]",
      "after": "derived = [[0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]",
      "relation": "implies",
      "rule_id": "Normality_ALL_g_LEFT_RIGHT_cosets",
      "conditions": [],
      "explanation": {
        "zh-Hant": "gH與Hg是同一g的完整左／右陪集；非交換群不能偷換乘法方向。",
        "en": "These are complete left and right cosets for the same g.Do not reverse multiplication in a nonabelian group."
      },
      "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": "S022",
      "uses": [
        "G001",
        "S021"
      ],
      "before": "source roles = [[0, 1, 2, 3, 4, 5, 6, 7], 是, 是, 是]",
      "after": "derived = [gHg^-1=H iff gH=Hg for EVERY g]",
      "relation": "implies",
      "rule_id": "Normality_GENERAL_conjugates_iff_left_right_cosets",
      "conditions": [],
      "explanation": {
        "zh-Hant": "右乘g是雙射，將共軛集合等式右乘g即得gH=Hg；反向右乘g^-1。此處不進一步宣稱quotient乘法良定。",
        "en": "Right multiplication by g is a bijection,turning gHg^-1=H into gH=Hg;right multiply by g inverse for the converse.No quotient multiplication is claimed here."
      },
      "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,
        1,
        2,
        3,
        4,
        5,
        6,
        7
      ],
      [
        [
          0,
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          true
        ],
        [
          1,
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          true
        ],
        [
          2,
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          true
        ],
        [
          3,
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          true
        ],
        [
          4,
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          true
        ],
        [
          5,
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          true
        ],
        [
          6,
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          true
        ],
        [
          7,
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          [
            0,
            1,
            2,
            3,
            4,
            5,
            6,
            7
          ],
          true
        ]
      ],
      true,
      "left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal"
    ]
  },
  "conclusion": {
    "canonical_value": "([0, 1, 2, 3, 4, 5, 6, 7], [[0, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [1, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [2, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [3, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [4, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [5, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [6, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是], [7, [0, 1, 2, 3, 4, 5, 6, 7], [0, 1, 2, 3, 4, 5, 6, 7], 是]], 是, left_and_right_cosets_agree_for_every_g_exactly_when_H_is_normal)",
    "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-20261113-r01-finite-normal-subgroup-foundations",
      "batch_path": "public/data/questions/batches/batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations/questions.jsonl",
      "batch_sha256": "2402d9b135e5666a3413fe9a70fcdfaf9cbdb5c57da7eb7fc407e30722481181",
      "original_line": 68,
      "original_line_sha256": "e1d2c2bfb7f9091aad08fcf0bbbf8fd6c8a611a1d15ad73c48e28a7a377efc01",
      "question_compact_sha256": "e7d578be62eb6fabd111abedea97d7b806d6c04ed77ea040db210d94ba89f115",
      "english_row_compact_sha256": "ae1ab26eadd9df2f96eb8563c4d65fa814994306cd6a382f11a60d3f371ee8c8",
      "english_manifest_path": "public/data/i18n/editions/batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261015-advance-20261113-r01-finite-normal-subgroup-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-finite-normal-subgroup": {
        "sha256": "65f7894921104c1d8e907fd0f2a4dfca9c6a56325d6cff31cbafd83e26dc5727",
        "english_sha256": "0416052f2e90eda2c0f56a07c41bbfb297a8c235ebb3ef3d41c72d7b4f26aade",
        "original_path": "public/data/mko/finite-normal-subgroup.json",
        "english_path": "public/data/i18n/editions/mko-finite-normal-subgroup.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "7f1649723787c6d7cf1abe85636fd3740f2c3cc1784071db7e38069361c81fd4",
    "original_candidate_sha256": "33974c0b54643a857e99e25d5e942ea8a15b2b8f546c2a6fb815b4a689c74a9a",
    "original_candidate_line_sha256": "33974c0b54643a857e99e25d5e942ea8a15b2b8f546c2a6fb815b4a689c74a9a"
  },
  "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-20261026-009-p49",
    "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": "049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0",
      "question_first_witness_sha256": "0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 22,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "bdd438bf21e2c267163ae9483a35863156f9f7fefcb9f4bd5a941da9d4c59fcc",
      "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."
  }
}