課程名稱︰代數導論二 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）︰2016/3/24 考試時限（分鐘）：120 試題 : Part I (30 points). True or False. Either prove the assertion or disprove it by a counter-example (6 point each): (1) A group of order 21 is not simple. (2) Every commutative group of order 8 is cyclic. (3) Every order 21 subgroup of the S_7 is commutative. (4) Every group of order 423 is simple. (5) The quotient G/Z(G), where Z(G) denotes the center of G, can not be cyclic, unless G is commutative. (6) The class equation of a group of order 2016 can be 1 + 1 + 2 + 3 + 4 + 31 + x. Part II (40 points). For each of the following problems, give accordingly a short proof or an example (8 point each): (1) Find two non-isomorphic non-commutative groups of order 8. (2) The class equation of a group of order 8 can not be 1 + 1 + 1 + 1 + 4. (3) If G and G' are both non-commutative groups of order 10, then G is isomorphic to G'. (4) If C(x) denotes the conjugacy class of some x in a group of order 55 such that |C(x)| = 11, then x is of order 5. (5) Let T be a Sylow 3-subgroup in a group G of order 105. If T is a normal subgroup of G, then T ⊂ Z(G). (6) If the class equation of G is 1 + 4 + 5 + 5 + 5, then G contains a normal Sylow 5-subgroup. Part III (30 points). Give a complete proof (10 points each): (1) Suppose a finite group G operates non-trivially on a finite set S of order r. If |G| > r!, then G is not simple. (2) A group of order p^2, p a prime number, is commutative. (3) Let p be a prime number and let G be a p-group. Let H be a proper subgroup of G. Prove that the normaliser of H is strictly larger than H. (4) Let G_1 ⊂ G be groups whose orders are divisible by p, and let H_1 be a Sylow p-subgroup of G_1. Proves that there exists a Sylow p-subgroup H of G such that H_1 = H ∩ G_1. --

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