{ "schema_version": "ocme-derivation-reader-v1", "guide_id": "derivation-mko-finite-group-center-01", "mko_id": "mko-finite-group-center", "title": { "zh-Hant": "全部交換rows認證群中心", "en": "Certifying the center from all commutation rows" }, "contract": { "schema_version": "ocme-derivation-guide-v0.1", "guide_id": "derivation-mko-finite-group-center-01", "source": { "mko_id": "mko-finite-group-center", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "sha256": "dd633886e48a7fb3d2d5a657d45487b7ff6cbff901ae305a94b8b1f8a966b58a", "english_sha256": "6d6ba95078de43aff968adedaeada2d4a1452d450419c12a944d6a75a98ee318" }, "mode": "worked_example", "audience_prerequisites": [ "calc" ], "goal": { "zh-Hant": "Z(G)={e}，直接驗subgroup及normal；G非abelian，r雖與自身交換卻不central。", "en": "Z(G)={e}, directly a normal subgroup; G is nonabelian and r is not central despite commuting with itself." }, "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", "S02" ], "before_tex": "z,g=e,r,r^2,s,rs,r^2s", "after_tex": "\\begin{array}{c|cccccc}gx=xg&e&r&r^2&s&rs&r^2s\\\\\\hline e&T&T&T&T&T&T\\\\r&T&T&T&F&F&F\\\\r^2&T&T&T&F&F&F\\\\s&T&F&F&T&F&F\\\\rs&T&F&F&F&T&F\\\\r^2s&T&F&F&F&F&T\\\\\\end{array}", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "完整36個boolean記錄zg=gz；每個row與全部六個g比較。", "en": "All 36 booleans record zg=gz, comparing each row with all six g values." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01", "S02" ], "explanation": { "zh-Hant": "T必來自同表兩個product相等，不是只看某一個g。", "en": "T comes from equality of two products in the same table, not a test against just one g." } }, "leaf_basis": "calc" }, { "id": "S04", "uses": [ "G1", "S03" ], "before_tex": "Z(G)=\\{z:\\ zg=gz\\ \\forall g\\in G\\}", "after_tex": "Z(G)=\\{e\\}", "relation": "implies", "rule": "mko_definition", "conditions": [], "explanation": { "zh-Hant": "只有e-row全T，其餘五row各有F，故完整中心只有identity。", "en": "Only e's row is all true; each other row contains false, so the full center contains only the identity." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S03" ], "explanation": { "zh-Hant": "r-row對e、r、r²為T仍不足，對s為F已否證central。", "en": "True entries against e,r,r2 are insufficient for r; the false entry against s refutes centrality." } }, "leaf_basis": "mko_definition" }, { "id": "S05", "uses": [ "G1", "S02", "S04" ], "before_tex": "Z=\\{e\\}", "after_tex": "e\\in Z,\\quad ee=e\\in Z,\\quad e^{-1}=e\\in Z\\Rightarrow Z\\le G", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "singleton載體的identity、全部一個product與全部一個inverse都直接核。", "en": "For this singleton carrier, identity, the sole product and the sole inverse are directly checked." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S02", "S04" ], "explanation": { "zh-Hant": "中心子群性不是從名稱跳過，完整有限責任已滿足。", "en": "Subgroup status is not skipped because of the name center; all finite obligations are met." } }, "leaf_basis": "calc" }, { "id": "S06", "uses": [ "G1", "S02", "S04", "S05" ], "before_tex": "g=e,r,r^2,s,rs,r^2s,\\quad z=e", "after_tex": "geg^{-1}=e,e,e,e,e,e\\text{ respectively}\\Rightarrow Z\\trianglelefteq G", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "全部六個conjugations保留唯一中心元素，直接證normal。", "en": "All six conjugations preserve the sole central element, directly proving normality." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S02", "S04", "S05" ], "explanation": { "zh-Hant": "normal與subgroup是兩層核對，前一步不能單獨代替此步。", "en": "Normality and subgroup status are separate checks; the preceding step cannot replace this one." } }, "leaf_basis": "calc" }, { "id": "S07", "uses": [ "G1", "S02", "S03", "S04" ], "before_tex": "gxg^{-1}=x\\ \\forall g", "after_tex": "gx=xg\\ \\forall g\\iff x\\in Z(G)\\iff\\operatorname{Cl}(x)=\\{x\\}", "relation": "equivalent", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "右乘g及反向右乘g⁻¹證singleton class與centrality等價；本例唯一e符合。", "en": "Multiplication by g and the reverse by g inverse prove singleton class iff centrality; only e qualifies here." }, "reverse_audit": { "kind": "inverse_equivalence", "required_ids": [ "G1", "S02", "S03", "S04" ], "explanation": { "zh-Hant": "不需要class-size公式或class equation，只用全稱交換定義。", "en": "No class-size formula or class equation is needed, only universal commutation." } }, "leaf_basis": "calc" }, { "id": "S08", "uses": [ "G1", "S01", "S03", "S04", "S07" ], "before_tex": "rs\\ne sr=r^2s", "after_tex": "G\\text{ not abelian};\\quad Z(G)\\ne G;\\quad Z(G)=G\\iff G\\text{ abelian}", "relation": "implies", "rule": "mko_definition", "conditions": [], "explanation": { "zh-Hant": "明示不交換pair否證abelian；一般等價是把全部z與g交換的同一句定義重寫。", "en": "An explicit noncommuting pair refutes abelianness; the general equivalence restates the same universal commutation condition." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01", "S03", "S04", "S07" ], "explanation": { "zh-Hant": "只與固定r交換的集合{e,r,r²}比中心大，不能把單元素centralizer當Z(G)。", "en": "The elements commuting with fixed r form the larger set {e,r,r2}; that single-element centralizer is not Z(G)." } }, "leaf_basis": "mko_definition" } ], "result": { "zh-Hant": "Z(G)={e}，直接驗subgroup及normal；G非abelian，r雖與自身交換卻不central。", "en": "Z(G)={e}, directly a normal subgroup; G is nonabelian and r is not central despite commuting with itself." }, "negative_controls": [ { "claim": { "zh-Hant": "r與r交換，所以r central。", "en": "r commutes with r, hence r is central." }, "expected": "reject", "reason": { "zh-Hant": "r與s不交換，central要求全部g。", "en": "r does not commute with s; centrality requires every g." } }, { "claim": { "zh-Hant": "Z(G)={e,r,r²}。", "en": "Z(G)={e,r,r2}." }, "expected": "reject", "reason": { "zh-Hant": "這是與r交換的集合；r、r²皆不與s交換。", "en": "That is the set commuting with r; r,r2 do not commute with s." } }, { "claim": { "zh-Hant": "群有限便abelian。", "en": "A finite group is abelian." }, "expected": "reject", "reason": { "zh-Hant": "此完整六元素表有rs≠sr。", "en": "This complete six-element table has rs distinct from sr." } } ], "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": "由完整 Cayley table 建立每個候選元素對所有 g 的 commutation row，Z(G) 是所有 row 全真者。需直接驗 identity、product、inverse closure 形成 subgroup，再對所有 g,z 驗 gzg^{-1} 仍在 Z(G) 以證 normal。z 的共軛類為單元素集，當且僅當 z∈Z(G)；Z(G)=G，當且僅當 G 為交換群。中心的定義要求與每個群元素交換，不能一般地用單一元素的 centralizer 取代；特定群中兩個集合可能相等。本輪不宣稱 characteristic 或 class equation。", "en": "From a complete qualified Cayley table, build each element z's commutation row against every g. The center Z(G) consists exactly of all-true rows. Directly check identity, product and inverse closure, then every conjugation to certify normality. z is central exactly when its full conjugacy class is a singleton, and Z(G)=G exactly when G is abelian. The center requires commutation with every group element, whereas a single-element centralizer imposes only that one commutation condition; they must not be identified in general, although they may coincide in particular groups. Characteristic status and the class equation are not asserted in this slice." }, "source": { "mko_id": "mko-finite-group-center", "sha256": "dd633886e48a7fb3d2d5a657d45487b7ff6cbff901ae305a94b8b1f8a966b58a", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f" } } ], "revision_log": [], "numeric_checks": [] }, "presentation": { "md_scope": { "zh-Hant": "完整六乘六commutation table、閉性與全共軛；不宣稱characteristic或class equation。", "en": "A complete six-by-six commutation table, closure and all conjugations; no characteristic or class-equation claim." }, "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" ], [ "T,F", "完整交換測試的真偽", "truth values of full commutation tests" ], [ "Z(G)", "所有全T rows的元素", "elements whose rows are all true" ] ], "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": { "id": "ocme-new142-reader-adoption-20261021", "collection": "coverage-20261021", "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "adopted_source_sha256": "a09d76035cc0fb9c7b94984a4710a247026139a690e8d946c8fb33de53d58b04", "original_source_sha256": "b8003d263c41d04f2bf8df3d9db33a054c738be3d16e50080408676978cb6c9f", "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": [ "mko_definition" ], "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW", "issue_ids": [ "BR142-P072", "BR142-P071" ], "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a", "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." } }