課程名稱︰代數導論一 課程性質︰數學系大二必修 課程教師︰莊武諺 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰102/11/07 考試時限（分鐘）：120 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : (1) (10 points) (The Third Isomorphism Theorem) Let G be a group and let H and K normal subgroups of G with H ≦ K. Prove that (G/H)/(K/H) is isomorphic to G/K, assuming the First Isomorphism theorem. (2) (10 points) Please classify all the groups with 4 elements. (Hint: Does the group contain an element of order 4?) (3) (10 points) Let G be a group. Prove that N = <(x^-1)(y^-1)xy|x,yεG> is a normal subgroup of G and G/N is abelian. (4) (10 points) Show that for any group G and any nonempty set A there is a bijection between the actions of G on A and the homomorphisms of G into Sym(A). (5) (10 points) Consider the group action of S3 on S3 itself by left multiplication. Re-label the elements of S3 by {_1 = (1)(2)(3), _2 = (1 2)(3), _3 = (1)(2 3), _4 = (1 3)(2), _5 = (1 2 3), _6 = (1 3 2)}. Then the bijection in Problem (4) give us a corresponding homomorphism Ψ:G → Sym(A), where G = S3 and A = S3. Please write down the images of _1,...,_6 under the homomorphism Ψ in the cycle decomposition form in Sym(A). (6) (5 points) Let G be a group. Show that a subgroup H of G with |G : H| = 2 is normal. (7) (5 points) Let ψ: G → H be a surjective group homomorphism and let N be a normal subgroup of G. Show that ψ(N) is a normal subgroup of H. (8) (5 points) Please give 3 different expression for the generator set of hte permutation group Sn, such that the order of the generator set is at most n-1. (i.e. Give the expressions for a finite set A such that Sn = <A> and |A| is at most n-1.) (9) (5 points) Show that S4 does not have a normal subgroup of order 8 or a normal subgroup of order 3. (10) (10 points) Let Φ: G → H be a group homomorphism and let N be a normal subgroup of G. Define a map ψ:G/N → H by ψ(gN) = Φ(g). Show that ψ is a well-defined group homomorphism if and only if ker(Φ) contains N. (11) (10 points) Let G acts on the set A. Show that if G acts transitively on A then the kernel of the action is given by ∩gεG gGg^-1 for any a ε A. (12) (10 points) Apply the class equation to prove that if p is a prime and P is a group of prime power order p^α for some α ≧ 1, then P has a nontrivial center. (13) (10 points) Let G be a finite group and let K and H be two normal subgroups of G such that G/H is isomorphic to K. Please show that if H and K are both simple, then G/K is isomorphic to H. (Recall that a group G is called simple if |G| ＞ 1 and the only normal subgroups of G are 1 and G.) (Hint: You could use the 2nd isomorphism theorem.) Remark: There are 110 points totally. 註：ε表屬於符號；S3表3個元素的對稱群 --

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