{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-finite-normal-subgroup-01",
  "mko_id": "mko-finite-normal-subgroup",
  "title": {
    "zh-Hant": "全部共軛認證rotation子群正規",
    "en": "Normality of rotations from every conjugate"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-finite-normal-subgroup-01",
    "source": {
      "mko_id": "mko-finite-normal-subgroup",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "65f7894921104c1d8e907fd0f2a4dfca9c6a56325d6cff31cbafd83e26dc5727",
      "english_sha256": "0416052f2e90eda2c0f56a07c41bbfb297a8c235ebb3ef3d41c72d7b4f26aade"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "calc"
    ],
    "goal": {
      "zh-Hant": "N={e,r,r²}為正規子群；H={e,s}雖為子群但不正規。",
      "en": "N={e,r,r2} is normal; H={e,s} is a subgroup but is not normal."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "G={e,r,r²,s,rs,r²s}，依序編碼(0,0),(1,0),(2,0),(0,1),(1,1),(2,1)。乘法(i,j)(k,l)=(i+(−1)^j k mod3,j+l mod2)。",
          "en": "G={e,r,r2,s,rs,r2s}, encoded respectively (0,0),(1,0),(2,0),(0,1),(1,1),(2,1), with (i,j)(k,l)=(i+(-1)^j*k mod3,j+l mod2)."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1"
        ],
        "before_tex": "G=\\{e,r,r^2,s,rs,r^2s\\}",
        "after_tex": "\\begin{array}{c|cccccc}*&e&r&r^2&s&rs&r^2s\\\\\\hline e&e&r&r^2&s&rs&r^2s\\\\r&r&r^2&e&rs&r^2s&s\\\\r^2&r^2&e&r&r^2s&s&rs\\\\s&s&r^2s&rs&e&r^2&r\\\\rs&rs&s&r^2s&r&e&r^2\\\\r^2s&r^2s&rs&s&r^2&r&e\\\\\\end{array}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "將給定配對公式逐36對展開，全部輸出在同一六元素載體，沒有缺格。",
          "en": "Expand the pair formula for all 36 products; every output belongs to the same six-element carrier and no cell is missing."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1"
          ],
          "explanation": {
            "zh-Hant": "labels依序綁定六個pair，故同名r²不代表額外第七元素。",
            "en": "The six labels bind the six pairs; r^2 is not an additional seventh element."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "(i,j)(k,l)=(i+(-1)^jk,j+l)\\quad(\\bmod(3,2))",
        "after_tex": "((i,j)(k,l))(m,n)=(i+(-1)^jk+(-1)^{j+l}m,j+l+n)=(i,j)((k,l)(m,n));\\quad e=(0,0);\\quad(i,j)^{-1}=(-(-1)^ji,j)",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩種括號的座標公式相同，證全部216 triples結合律；(0,0)兩側中性，列出的逆元兩側乘回(0,0)。",
          "en": "Both bracketings have identical coordinates, proving associativity for all 216 triples; (0,0) is a two-sided identity and the inverse formula multiplies back on both sides."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "指數j+l只依mod2，因(−1)²=1，公式對所有代表一致；逆元清單e,r²,r,s,rs,r²s完整。",
            "en": "The sign depends only on j+l modulo 2 since (-1)^2=1; the full inverse list is e,r^2,r,s,rs,r^2s."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S01"
        ],
        "before_tex": "N=\\{e,r,r^2\\}",
        "after_tex": "\\begin{array}{c|ccc}*&e&r&r^2\\\\\\hline e&e&r&r^2\\\\r&r&r^2&e\\\\r^2&r^2&e&r\\\\\\end{array}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "N全部九乘積在N，且e∈N。",
          "en": "All nine products in N remain in N, and e belongs to N."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01"
          ],
          "explanation": {
            "zh-Hant": "subgroup必先核closure，大小3本身不足。",
            "en": "Subgroup closure must be checked first; size 3 alone is insufficient."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S02",
          "S03"
        ],
        "before_tex": "e^{-1}=e,\\quad r^{-1}=r^2,\\quad(r^2)^{-1}=r",
        "after_tex": "N^{-1}=N\\Rightarrow N\\le G",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "全部三個逆元在N，完成子群資格。",
          "en": "All three inverses lie in N, completing subgroup qualification."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "正規性尚未證，不能因有限或index數字直接升格。",
            "en": "Normality has not yet been proved; finiteness or an index number alone is not used to upgrade it."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S01",
          "S02",
          "S04"
        ],
        "before_tex": "g=e,r,r^2,s,rs,r^2s",
        "after_tex": "\\begin{array}{c|ccc}gng^{-1}&e&r&r^2\\\\\\hline e&e&r&r^2\\\\r&e&r&r^2\\\\r^2&e&r&r^2\\\\s&e&r^2&r\\\\rs&e&r^2&r\\\\r^2s&e&r^2&r\\\\\\end{array}",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整18格共軛表：rotations固定r、r²，reflections交換r與r²，每row的集合皆N。",
          "en": "The full 18-entry conjugation table fixes r,r^2 for rotations and swaps them for reflections; every row set equals N."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S02",
            "S04"
          ],
          "explanation": {
            "zh-Hant": "所有六個g均核，沒有只測r、s而省略生成責任。",
            "en": "All six g values are checked rather than testing r,s without a generation proof."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S01",
          "S04",
          "S05"
        ],
        "before_tex": "N=\\{e,r,r^2\\},\\quad B=\\{s,rs,r^2s\\}",
        "after_tex": "gN=Ng=N\\ (g=e,r,r^2);\\qquad gN=Ng=B\\ (g=s,rs,r^2s)",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整左右cosets也逐六元素核相等，提供正規性的另一等價檢查。",
          "en": "Left and right cosets agree for each of the six elements, providing another normality check."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S04",
            "S05"
          ],
          "explanation": {
            "zh-Hant": "這裡cosets值由父表所有三元素乘積取得，不是代表元單一像。",
            "en": "Each coset uses all three parent-table products, not a single representative image."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S04",
          "S05",
          "S06"
        ],
        "before_tex": "N\\le G,\\quad\\forall g:\\ gNg^{-1}=N",
        "after_tex": "N\\trianglelefteq G",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "子群與全稱共軛責任都完成，故N正規。",
          "en": "Both subgroup and universal conjugation obligations are complete, so N is normal."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "S05",
            "S06"
          ],
          "explanation": {
            "zh-Hant": "未在此宣稱任何商群表或第一同構定理。",
            "en": "No quotient table or first-isomorphism theorem is claimed here."
          }
        },
        "leaf_basis": "mko_definition"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S01",
          "S02"
        ],
        "before_tex": "H=\\{e,s\\}",
        "after_tex": "ee=e,es=s,se=s,ss=e;\\quad e^{-1}=e,s^{-1}=s\\Rightarrow H\\le G",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "另一candidate H先完整核四乘積與兩逆元，確實也是子群。",
          "en": "First check all four products and two inverses for H, establishing it is a subgroup too."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "後面的反例否證正規，不是否證H的子群性。",
            "en": "The upcoming witness refutes normality, not H's subgroup status."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S09",
        "uses": [
          "G1",
          "S01",
          "S02",
          "S08"
        ],
        "before_tex": "g=r,\\quad h=s\\in H",
        "after_tex": "r s r^{-1}=r^2s\\notin H\\Rightarrow H\\not\\trianglelefteq G",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "一個明示escaping conjugate已足否證全稱normality。",
          "en": "One explicit escaping conjugate suffices to refute universal normality."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "S02",
            "S08"
          ],
          "explanation": {
            "zh-Hant": "r⁻¹=r²由同表取得；乘法順序不可反轉。",
            "en": "r inverse is r^2 from the same table; product order cannot be reversed."
          }
        },
        "leaf_basis": "calc"
      },
      {
        "id": "S10",
        "uses": [
          "G1",
          "S02",
          "S07",
          "S09"
        ],
        "before_tex": "g\\in G",
        "after_tex": "g\\{e\\}g^{-1}=\\{e\\},\\quad gGg^{-1}=G",
        "relation": "implies",
        "rule": "calc",
        "conditions": [],
        "explanation": {
          "zh-Hant": "trivial與whole subgroup皆正規；後者共軛映射有逆g⁻¹的共軛而為雙射。",
          "en": "The trivial and whole subgroups are normal; conjugation of G is onto because its inverse is conjugation by g inverse."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "S07",
            "S09"
          ],
          "explanation": {
            "zh-Hant": "這兩個邊界不使其他任意subgroup也正規。",
            "en": "These boundaries do not make every other subgroup normal."
          }
        },
        "leaf_basis": "calc"
      }
    ],
    "result": {
      "zh-Hant": "N={e,r,r²}為正規子群；H={e,s}雖為子群但不正規。",
      "en": "N={e,r,r2} is normal; H={e,s} is a subgroup but is not normal."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "H大小2整除6故正規。",
          "en": "H has size 2 dividing 6, hence normal."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "r s r⁻¹=r²s不在H。",
          "en": "r*s*r inverse=r^2s lies outside H."
        }
      },
      {
        "claim": {
          "zh-Hant": "只看N的乘積封閉便已證正規。",
          "en": "Product closure of N alone proves normality."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "閉性只供子群責任，還需全部g共軛；H就是子群但不正規反例。",
          "en": "Closure addresses subgroup status; all conjugations remain necessary, as subgroup H is not normal."
        }
      },
      {
        "claim": {
          "zh-Hant": "共軛只測identity就足。",
          "en": "Testing conjugation by identity suffices."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "identity共軛對任何subset都不改值，不能排除H的r反例。",
          "en": "Identity conjugation fixes every subset and cannot detect H's r witness."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "calc",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "假設精確模3、模2整數算術、有限集合外延相等與群／作用的定義；不預設此運算已為群。",
          "en": "Assume exact arithmetic modulo 3 and 2, extensional finite-set equality, and definitions of groups/actions; this operation is not pre-assumed to be a group."
        }
      },
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "G須由完整Cayley table認證，H須先完整通過identity、product與inverse closure。正規性要求對每個g∈G完整驗gHg^{-1}=H；等價條件是每個g皆有gH=Hg。一個明示g,h使ghg^{-1}不屬H即可否證normal。trivial subgroup與whole group均正規。有限性本身、抽樣共軛或只檢生成元但未證生成責任都不足；本節點尚不宣稱quotient multiplication well-defined。",
          "en": "The complete Cayley table must certify G, and H must first pass identity, product and inverse closure. Normality then requires gHg^(-1)=H for every g in G, equivalently gH=Hg for every g. One explicit g,h with h in H but g*h*g^(-1) outside H refutes normality. Trivial and whole subgroups are normal. Finiteness, sampled conjugates or generator checks without a proved generation obligation are insufficient. Well-defined quotient multiplication is not yet claimed here."
        },
        "source": {
          "mko_id": "mko-finite-normal-subgroup",
          "sha256": "65f7894921104c1d8e907fd0f2a4dfca9c6a56325d6cff31cbafd83e26dc5727",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": []
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "完整subgroup與所有g共軛／左右cosets核對，不在此建立商乘法。",
      "en": "Complete subgroup, all-g conjugation and left/right coset checks; quotient multiplication is not constructed here."
    },
    "md_symbols": [
      [
        "G",
        "六元素群載體",
        "six-element group carrier"
      ],
      [
        "r,s",
        "r³=e,s²=e,sr=r²s的labels",
        "labels with r^3=e,s^2=e,sr=r^2s"
      ],
      [
        "N,H",
        "正規與非正規子群",
        "normal and nonnormal subgroups"
      ]
    ],
    "md_error_controls": [
      {
        "title": "同一群表 / Same group table",
        "zh": "全部積、逆元與cosets綁定這個完整Cayley表。",
        "en": "Bind every product, inverse and coset to this complete Cayley table."
      },
      {
        "title": "完整枚舉 / Complete enumeration",
        "zh": "任何全稱有限性質都需全載體或全稱公式，不用proper sample。",
        "en": "Every universal finite property needs the full carrier or a universal formula, not a proper sample."
      }
    ],
    "md_open_items": []
  },
  "revision": {
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    "collection": "coverage-20261021",
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    "original_source_sha256": "afda0b4ee1c3617c0aaa2344f6db7818aaff3418cd954b96fa712b07765a328e",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
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    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}