{ "schema_version": "ocme-derivation-reader-v1", "guide_id": "derivation-mko-finite-group-conjugacy-class-01", "mko_id": "mko-finite-group-conjugacy-class", "title": { "zh-Hant": "完整共軛作用與三個classes", "en": "Full conjugation action and three classes" }, "contract": { "schema_version": "ocme-derivation-guide-v0.1", "guide_id": "derivation-mko-finite-group-conjugacy-class-01", "source": { "mko_id": "mko-finite-group-conjugacy-class", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "sha256": "64bbc8cfc6df1fef690cba391584bc0ae2770f3af76115c24634669363f3bb26", "english_sha256": "026cb09f3fdfee6e3b2779e163410961306f437b84d80dc9b8673be1d56d9511" }, "mode": "worked_example", "audience_prerequisites": [ "calc" ], "goal": { "zh-Hant": "全部共軛類是{e}、{r,r²}、{s,rs,r²s}；r與s不共軛。", "en": "The conjugacy classes are {e},{r,r2},{s,rs,r2s}; r and s are not conjugate." }, "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": "g,x\\in G,\\quad c_g(x)=gxg^{-1}", "after_tex": "\\begin{array}{c|cccccc}gxg^{-1}&e&r&r^2&s&rs&r^2s\\\\\\hline e&e&r&r^2&s&rs&r^2s\\\\r&e&r&r^2&r^2s&s&rs\\\\r^2&e&r&r^2&rs&r^2s&s\\\\s&e&r^2&r&s&r^2s&rs\\\\rs&e&r^2&r&r^2s&rs&s\\\\r^2s&e&r^2&r&rs&s&r^2s\\\\\\end{array}", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "由父表及全部逆元計出完整36格conjugation table，載體仍G。", "en": "Compute every one of the 36 conjugation entries from the parent table and all inverses; the point carrier is G itself." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01", "S02" ], "explanation": { "zh-Hant": "每列g、每column x的角色明示，不把subgroup conjugate gHg⁻¹混入。", "en": "Rows index g and columns x, not conjugate subgroups gHg inverse." } }, "leaf_basis": "calc" }, { "id": "S04", "uses": [ "G1", "S02", "S03" ], "before_tex": "c_e(x)=exe^{-1}=x", "after_tex": "c_{gh}(x)=(gh)x(h^{-1}g^{-1})=g(hxh^{-1})g^{-1}=c_g(c_h(x))", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "群律證單位與全部composition相容，所以這是已認證的共軛作用。", "en": "Group laws prove identity and composition for every index, qualifying the conjugation action." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S02", "S03" ], "explanation": { "zh-Hant": "(gh)⁻¹=h⁻¹g⁻¹需反序，否則相容式會錯。", "en": "The inverse of gh is h inverse g inverse in reverse order; the wrong order would invalidate compatibility." } }, "leaf_basis": "calc" }, { "id": "S05", "uses": [ "G1", "S03", "S04" ], "before_tex": "\\operatorname{Cl}(x)=\\{c_g(x):g\\in G\\}", "after_tex": "\\operatorname{Cl}(e)=\\{e\\},\\quad\\operatorname{Cl}(r)=\\operatorname{Cl}(r^2)=\\{r,r^2\\}", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "完整e、r、r² columns去重得到identity類與rotation類。", "en": "Deduplicate the complete columns for e,r,r2 to obtain the identity and rotation classes." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S03", "S04" ], "explanation": { "zh-Hant": "未只看r自己共軛自己，還包含全部reflections給的r²。", "en": "The calculation includes all reflections producing r2, not merely r conjugating itself." } }, "leaf_basis": "calc" }, { "id": "S06", "uses": [ "G1", "S03", "S04", "S05" ], "before_tex": "x=s,rs,r^2s", "after_tex": "\\operatorname{Cl}(s)=\\operatorname{Cl}(rs)=\\operatorname{Cl}(r^2s)=\\{s,rs,r^2s\\}", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "其餘三columns完整遍歷同一reflection集合，全部六個元素的classes都已列。", "en": "The remaining three full columns each traverse the reflection set, completing all six elements' classes." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S03", "S04", "S05" ], "explanation": { "zh-Hant": "多個元素共享同一class，去重後只有一個reflection類。", "en": "Several elements share one class; deduplication leaves one reflection class." } }, "leaf_basis": "calc" }, { "id": "S07", "uses": [ "G1", "S05", "S06" ], "before_tex": "C_0=\\{e\\},C_1=\\{r,r^2\\},C_2=\\{s,rs,r^2s\\}", "after_tex": "C_i\\cap C_j=\\varnothing\\ (i\\ne j),\\quad C_0\\cup C_1\\cup C_2=G", "relation": "implies", "rule": "mko_definition", "conditions": [], "explanation": { "zh-Hant": "三個互異classes直接互斥覆蓋，亦符合群作用orbits相等或不交。", "en": "The three distinct classes directly form a disjoint cover, agreeing with the equal-or-disjoint orbit property." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S05", "S06" ], "explanation": { "zh-Hant": "這是classes的partition，不使用任何centralizer計數。", "en": "This is a class partition without a centralizer count." } }, "leaf_basis": "mko_definition" }, { "id": "S08", "uses": [ "G1", "S02", "S05", "S06", "S07" ], "before_tex": "\\operatorname{Cl}(x)=\\{x\\}", "after_tex": "gxg^{-1}=x\\ \\forall g\\iff gx=xg\\ \\forall g", "relation": "equivalent", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "右乘g與反向右乘g⁻¹證singleton恰當與每個g交換；本例只有e類singleton。", "en": "Right multiplication by g and the reverse by g inverse show singleton exactly when x commutes with every g; here only e has a singleton class." }, "reverse_audit": { "kind": "inverse_equivalence", "required_ids": [ "G1", "S02", "S05", "S06", "S07" ], "explanation": { "zh-Hant": "這只用共軛定義，不引入class-size或其他後續定理。", "en": "This uses only conjugation, not a class-size formula or later theorem." } }, "leaf_basis": "calc" }, { "id": "S09", "uses": [ "G1", "S03", "S05", "S07" ], "before_tex": "s\\notin\\operatorname{Cl}(r)=\\{r,r^2\\}", "after_tex": "r\\text{ and }s\\text{ nonconjugate}", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "完整r-column未含s，已排除所有g，認證nonconjugacy。", "en": "The full r-column excludes s, exhausting all g and certifying nonconjugacy." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S03", "S05", "S07" ], "explanation": { "zh-Hant": "一個g不能送r到s不足否證；這裡使用完整column。", "en": "Failure for one g would not suffice; the complete column is used." } }, "leaf_basis": "calc" } ], "result": { "zh-Hant": "全部共軛類是{e}、{r,r²}、{s,rs,r²s}；r與s不共軛。", "en": "The conjugacy classes are {e},{r,r2},{s,rs,r2s}; r and s are not conjugate." }, "negative_controls": [ { "claim": { "zh-Hant": "r的class只有{r}。", "en": "The class of r is only {r}." }, "expected": "reject", "reason": { "zh-Hant": "srs⁻¹=r²，亦在class。", "en": "s*r*s inverse=r2 also belongs to the class." } }, { "claim": { "zh-Hant": "r與s共軛因同屬G。", "en": "r and s are conjugate because both lie in G." }, "expected": "reject", "reason": { "zh-Hant": "完整Cl(r)只有r、r²，不含s。", "en": "The full class of r contains only r,r2, not s." } }, { "claim": { "zh-Hant": "每個conjugacy class都是subgroup。", "en": "Every conjugacy class is a subgroup." }, "expected": "reject", "reason": { "zh-Hant": "{r,r²}不含identity e。", "en": "{r,r2} omits the identity e." } } ], "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": "必須從完整 Cayley table 與 inverse 建立完整 conjugation action table，逐一驗 identity/action composition。指定 x 的 class 是所有 gxg^{-1} 的完整集合；全部 classes 兩兩相等或不交且 union 為 G。singleton class 恰表示 x 與所有 g commute。明示 y 不在 class(x) 即可證兩元素 nonconjugate。共軛類、coset 與 conjugate subgroup 是不同構造，不應混同；特定情況下其底層集合可以恰好相等。本輪不使用 class-size=[G:C_G(x)]，因此不引入 centralizer 或額外 orbit-stabilizer 結論。", "en": "Derive the full conjugation table from the complete group table and inverses, checking identity and action composition. The class of x contains every g*x*g^(-1), and all classes are equal or disjoint and cover G. A singleton class means x commutes with every group element; an explicit y outside the class proves nonconjugacy. Conjugacy classes, cosets and conjugate subgroups are different constructions and must not be conflated, even when particular underlying sets coincide. No centralizer, class-size formula or additional orbit-stabilizer consequence is used here." }, "source": { "mko_id": "mko-finite-group-conjugacy-class", "sha256": "64bbc8cfc6df1fef690cba391584bc0ae2770f3af76115c24634669363f3bb26", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f" } } ], "revision_log": [], "numeric_checks": [] }, "presentation": { "md_scope": { "zh-Hant": "完整Cayley導出共軛action、全部orbits與partition；不用centralizer或class-size公式。", "en": "Derive complete conjugation, orbits and partition from Cayley data without centralizers or a class-size formula." }, "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" ], [ "\\operatorname{Cl}(x)", "element x的共軛類", "conjugacy class of element x" ] ], "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": "18e9a1b4f67b6f1182f4512c4416da956b0d05cdac22fa05cd0d5f38ac074087", "original_source_sha256": "a76c966c381c514457e23c2308b223bd35aee8d53f4c9b6b577beb4dce19d883", "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-P067", "BR142-P066" ], "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." } }