{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","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-group"},"kind":"guides","total":7,"pages":1,"page":1,"entries":[{"id":"derivation-mko-finite-group-01","title":{"zh-Hant":"有限群：以完整 Cayley 表核對群公理","en":"Finite group: auditing the group axioms from a complete Cayley table"},"summary":{"zh-Hant":"給兩個完整運算表。（甲）G={0,1,2,3} 配上表 T，證明 (G,*) 是有限群：封閉、全部 4³=64 個三元組的結合律、雙側單位元、雙側逆元。（乙）K={0,1,2} 配上表 ∘：它對稱、有雙側單位元、每個元素都有雙側逆元，卻在三元組 (1,1,2) 違反結合律，因此不是群；同時說明含 0 的 19 個三元組全部通過，抽樣會誤判。","en":"Two complete operation tables are given. (A) For G={0,1,2,3} with table T, prove that (G,*) is a finite group: closure, associativity for all 4³=64 triples, a two-sided identity and two-sided inverses. (B) K={0,1,2} with table ∘ is symmetric, has a two-sided identity and two-sided inverses for every element, yet violates associativity at the triple (1,1,2), so it is not a group; also show that all 19 triples containing 0 pass, so sampling would misjudge it."},"mko_ids":["mko-finite-group"]},{"id":"derivation-mko-finite-group-action-01","title":{"zh-Hant":"六元素群對四點的完整作用","en":"A complete six-element action on four points"},"summary":{"zh-Hant":"明示24格表為有效群作用，但不是free或transitive。","en":"The explicit 24-entry table is a valid group action, but it is neither free nor transitive."},"mko_ids":["mko-finite-group-action"]},{"id":"derivation-mko-finite-group-orbit-01","title":{"zh-Hant":"完整orbits與partition","en":"Complete orbits and their partition"},"summary":{"zh-Hant":"X的全部互異orbits為{0,1,2}與{3}，同orbit關係是等價關係。","en":"The distinct orbits are {0,1,2} and {3}; same-orbit is an equivalence relation."},"mko_ids":["mko-finite-group-orbit"]},{"id":"derivation-mko-finite-group-stabilizer-01","title":{"zh-Hant":"point0的完整固定群元素","en":"All group elements fixing point zero"},"summary":{"zh-Hant":"Stab_G(0)={e,s}為子群；Stab_G(3)=G，與orbit及action kernel不同。","en":"Stab_G(0)={e,s} is a subgroup; Stab_G(3)=G, distinct from orbit and action-kernel constructions."},"mko_ids":["mko-finite-group-stabilizer"]},{"id":"derivation-mko-finite-group-homomorphism-kernel-01","title":{"zh-Hant":"C₄到C₂的kernel、fibers與單射性","en":"Kernel, fibers and injectivity from C4 to C2"},"summary":{"zh-Hant":"f:C₄→C₂，f(a)=a mod2的kernel為{0,2}且正規；fibers是其cosets，f非單射。","en":"For f:C4 to C2, f(a)=a mod 2, the normal kernel is {0,2}; fibers are its cosets and f is not injective."},"mko_ids":["mko-finite-group-homomorphism-kernel"]},{"id":"derivation-mko-finite-group-conjugacy-class-01","title":{"zh-Hant":"完整共軛作用與三個classes","en":"Full conjugation action and three classes"},"summary":{"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."},"mko_ids":["mko-finite-group-conjugacy-class"]},{"id":"derivation-mko-finite-group-center-01","title":{"zh-Hant":"全部交換rows認證群中心","en":"Certifying the center from all commutation rows"},"summary":{"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."},"mko_ids":["mko-finite-group-center"]}],"title":"概念推導導讀"}