{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-finite-abelian-group"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-finite-abelian-group-01","title":{"zh-Hant":"有限阿貝爾群：由完整 Cayley 表先核群資格、再核交換性","en":"Finite abelian group: from a complete Cayley table, group qualification first, then commutativity"},"summary":{"zh-Hant":"載體 G={0,1,2,3} 上給出完整的 4×4 Cayley 表（行 a、列 b 的格子是 a*b）：第 0 行 (1,2,3,0)，第 1 行 (2,3,0,1)，第 2 行 (3,0,1,2)，第 3 行 (0,1,2,3)。依序核對封閉、結合、唯一單位元與逆元，再對所有 a、b 核對 a*b=b*a，判定 (G,*) 是否為有限阿貝爾群；求元素 0 與 1 的階與它們生成的循環子群；並以 {0,1,2} 上的對稱表 a∘b=|a−b| 說明「表對稱」不能代替群資格。","en":"On the carrier G={0,1,2,3} a complete 4×4 Cayley table is given (the cell in row a, column b is a*b): row 0 is (1,2,3,0), row 1 is (2,3,0,1), row 2 is (3,0,1,2), row 3 is (0,1,2,3). Check closure, associativity, a unique identity and inverses in that order, then check a*b=b*a for all a, b, to decide whether (G,*) is a finite abelian group; find the orders of the elements 0 and 1 and the cyclic subgroups they generate; and use the symmetric table a∘b=|a−b| on {0,1,2} to show that a symmetric table cannot replace group qualification."},"mko_ids":["mko-finite-abelian-group"]}],"title":"Concept derivation guides"}