課程名稱︰代數導論一 課程性質︰必修 課程教師︰陳其誠 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2011/11/24 考試時限（分鐘）：190 mins 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Write your answer on the answer sheet. We give partial points. In this examina- tions, G denotes a group, H, K, denote subgroups of G, and e always is the ide- ntity of G. Also, Α(G) denote the group of all automorphisms of G, and N(H) = -1 { x∈G | xHx = H } (1) (5 points each) "yes" or "no". Either give a brief reason or give a counter example. (a) If both H and K are normal subgroups of G with H∩K = (e), then H com- mutes with K. (b) The order of an element of the symmetric group S is at most 7. 7 (c) The alternating group A is simple. 3 (d) The alternating group A is simple. 4 (e) The number of conjugates of (1,2,3) in S equals 20. 5 (f) If o(G) is an even natural number, then there exist some x∈G, x≠e such that the number of conjugates of x in G is odd. (2) (15 points) Suppose H is cyclic of order 9. (a) Prove that Α(H) is of order 6 (10 points). (b) Prove that if K is a finite group of order relatively prime to 6, then K commutes with H (5 points). (3) (20 points) Suppose o(G) = pq where p and q are prime numbers with p > q and let P and Q are respectively p-Sylow and q-Sylow subgroups of G. (a) Show that P is normal in G (5 points). (b) Show that if p≠1 (mod q), then Q is normal in G (5 points). (↑是"≡"同餘的否定，因為符號打不出來) (c) Show that if o(G) = 35, then G is cyclic (10 points). (4) (10 points) Show that if H is a p-Sylow subgroup of G, then N(N(H)) = N(H). (5) (a) Prove that for n≧3, the subgroup of S generated by all 3-cycles n in A (10 points). n (b) Prove if G = S and H is a normal subgroup of G containing a 3-cycle, n then H contains A (5 points). n (6) (10 points) If o(G) = p^n where p is a prime number, prove that there exist subgroups N , i = 0, 1, ..., r, for some r, such that G = N ⊃= N ⊃= ... i 0 1 (⊃=：包含或等於) ⊃= N = (e) where N is a normal subgroup of N and where N /N is r i i-1 i-1 i abelian. --

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