{ 萬年考古題 ＸＤ } 定義部份+解答 1.semi-group(半群): {G,0} is called a semi-group. if for all a,b belong to G => (a。b)。c = a。(b。c) 2.group(群) {G,0} is called a group. <1> for all a.b belong to G => a。b also belong to G. <2> for all a.b.cbelong to G => (a。b)。c = a。(b。c) <3> exist e belong to G,and for all a belong to G => e。a = e = a。e <4> for all a belonf to G, and exist a。a^(-1) = a^(-1)。a = e 3.abelian(commutative) group(交換群) a group G is called ab abelian group , if for all a.b belong to G => a。b = b。a 4.order(秩) G be a group is called the order of G is the number of element of element and denoted by 。(G) . [or |G|] 5.symmetric group (對稱群) The symmtric group of degree n is the set of all 1-1 , mappings of a set n element onto itself and denoted by Sn , then clearly: 。(Sn) = n！ 6.subgroup(子群) G is a group H≦G , then H is called subgroup ,if H itself is (包含於) a group & denoted by H < G. 7.cyclic group(循環群) A group G is called cyclic group generated by g ,if exists g belong to G , such that G={g^n|n belong to Z} and denoted by G=<g>. 8.coset(伴集) If H<G , a belong to G ,then: ∕Ha = {ha|h belong to G} is called a right coset of H by a. ﹨aH = {ah|h brlong to G} is called a left coset of H by a. 9.index(指標) If H<G ,the index of H in G is the number of distinct right(left) cosets of H in G and denoted by indG(H) (ie. G is a finite group ,then indG(H)= 。(G) ) ─── 。(h) 10.Largrage Theorem If G is a finite froup and H is a subgroup of G , then 。(H)|。(G). (。(H) is a divisor of 。(G) ). 11.normal subgroup(正規子群) H<G ,H is said to be a normal subgroup of G, if for all g belong to H, g belong to G => g^(-1)hg belong to H and denoted by H△G (△一個角是朝左,好奇怪,打不出來,符號一直變問號) 12.Quotient group(商群) H△G (同上一題,一個角朝左,像小於的形狀), a set G/H = {Hg|g belong to G} is called a quotient of G by H . 13.center G is a group. Z(G)={ Z belong to G| ZX=XZ, for all X belong to G } is called the center(or zenter) of G. 14.normalizer H<G, N(H) = {Hg=gH| g belong to G} is called the normalizer of a in G. 15.centralizer G is a group, a belong to G, Ca = {x|ax=xa , x belong to a} is called the centralizer of a. 16.homomorphism(同態函數) {G,0},{G,*} are two groups, if for all x,y belong to G => f(x。y) = f(x)*f(y) , then f is called a homomorphism from G into H. 17.isomorphism(同構函數) A homomorphism f is called isomorphism if there f is one-to-one . 18.isomorphic(同構) {G,0},{H,*} are two groups ,G.H are called isomorphic if there is an isomorphism from G into H . 19.permutation(重排) A bijection(1-1&onto) from a set S into itself is called a permution of S. 20.Cayley's Theorem Every group is isomorphic to a permutation froup. 21.simple A group is said to be simple if it has no homomorphic image. (ie.if it has no nontrivial normal subgroups) 22.First Isomorphism Theorem G.H two groups. δ: G→H (homomorphism) with Kerδ=K, then G/K=H . 23.alternating group(變換群) An is called the alternating group of degree n, then An is the set of all even permutation of Sn . 24.Conjugate(共軛) G is a group a.b belong to G, then a.b is called conjugates if there exist c belong to G such that b = c^(-1)ac ,and denoted by a～b . 25.class equation If G is a finite group, then Ca = 。(G)/。(N(a)) , (ie.the number of lelments conjugate to a in G is the index of the normalizer of a in G) 。(G) => 。(G) = Σ ──── 。(N(a)) --

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