{ "schema_version": "ocme-derivation-reader-v1", "guide_id": "derivation-mko-cauchy-theorem-finite-groups-01", "mko_id": "mko-cauchy-theorem-finite-groups", "title": { "zh-Hant": "prime因子的exact-order witnesses", "en": "Exact-order witnesses for the prime divisors" }, "contract": { "schema_version": "ocme-derivation-guide-v0.1", "guide_id": "derivation-mko-cauchy-theorem-finite-groups-01", "source": { "mko_id": "mko-cauchy-theorem-finite-groups", "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f", "sha256": "779e8ef831907c123fc1e7039941b328b38b54af646d6a171eab7e28b476927d", "english_sha256": "557db8e818ba9af78d5bbcd15fb786238f5f2099b296f105fa26a68e62fd821f" }, "mode": "worked_example", "audience_prerequisites": [ "calc" ], "goal": { "zh-Hant": "此六元素群的prime divisors2、3分別有s與r的exact-order witnesses；無order6元素。", "en": "In this six-element group, prime divisors 2,3 have exact-order witnesses s,r; there is no element of order 6." }, "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|=6", "after_tex": "6=2\\cdot3,\\quad2,3\\text{ prime},\\quad2\\mid6,3\\mid6", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "完整群有六元素，prime divisor只有2與3，兩個均符合定理前提。", "en": "The qualified group has six elements and prime divisors exactly 2 and 3, both meeting the theorem premise." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01", "S02" ], "explanation": { "zh-Hant": "6也整除6但不是prime，不能把它當Cauchy的p。", "en": "6 divides 6 but is not prime, so it cannot be the theorem's p." } }, "leaf_basis": "calc" }, { "id": "S04", "uses": [ "G1", "S01", "S03" ], "before_tex": "p=2,\\quad g=s", "after_tex": "s^1=s\\ne e,\\quad s^2=e\\Rightarrow\\operatorname{ord}(s)=2", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "列出全部較小正冪k=1及第2冪回返，得到exact2而非僅g²=e。", "en": "Check the only smaller positive exponent 1 and return at exponent 2, proving exact order 2 rather than merely g^2=e." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01", "S03" ], "explanation": { "zh-Hant": "s與e是不同table labels，排除了order1。", "en": "s and e are distinct table labels, excluding order 1." } }, "leaf_basis": "calc" }, { "id": "S05", "uses": [ "G1", "S01", "S03" ], "before_tex": "p=3,\\quad g=r", "after_tex": "r^1=r\\ne e,\\quad r^2\\ne e,\\quad r^3=e\\Rightarrow\\operatorname{ord}(r)=3", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "兩個全部較小正冪都非identity，第3冪回返，得到exact3。", "en": "Both smaller positive powers are nonidentity and the third returns, proving exact order 3." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01", "S03" ], "explanation": { "zh-Hant": "只寫r³=e不夠；e也滿足同式卻只有order1。", "en": "r^3=e alone is insufficient: e also satisfies it but has order 1." } }, "leaf_basis": "calc" }, { "id": "S06", "uses": [ "G1", "S01", "S02", "S04", "S05" ], "before_tex": "g=e,r,r^2,s,rs,r^2s", "after_tex": "\\operatorname{ord}(g)=1,3,3,2,2,2\\text{ respectively}", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "r²的冪為r²,r,e；三個reflection各第1冪非e、第2冪e，完整六項階表。", "en": "Powers of r^2 are r^2,r,e; each reflection is nonidentity at power 1 and identity at power 2, completing all six orders." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S01", "S02", "S04", "S05" ], "explanation": { "zh-Hant": "每個元素都出現一次，非只測生成元r、s。", "en": "Every element appears once, not just the generators r,s." } }, "leaf_basis": "calc" }, { "id": "S07", "uses": [ "G1", "S03", "S06" ], "before_tex": "6\\mid|G|,\\quad6\\notin\\{1,2,3\\}", "after_tex": "\\nexists g\\in G:\\operatorname{ord}(g)=6", "relation": "implies", "rule": "calc", "conditions": [], "explanation": { "zh-Hant": "完整階表排除composite divisor6的元素，展示prime結論不可擴成任意divisor。", "en": "The complete order table excludes an element of composite order 6, showing that the prime conclusion does not extend to every divisor." }, "reverse_audit": { "kind": "premise_audit", "required_ids": [ "G1", "S03", "S06" ], "explanation": { "zh-Hant": "這個不存在性由六元素完整窮舉成立，不是假借Cauchy converse。", "en": "Nonexistence follows from exhaustive enumeration, not a supposed Cauchy converse." } }, "leaf_basis": "calc" }, { "id": "S08", "uses": [ "G1", "S03", "S04", "S05", "S06", "S07" ], "before_tex": "p\\in\\{2,3\\}\\text{ prime divisor of }6", "after_tex": "\\exists g\\in G:\\ g^p=e\\text{ and }g^k\\ne e\\ (1\\le k