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）︰2015/10/22 考試時限（分鐘）：130(110+20) 試題 : Write your answer on the answer sheet. You should include in your answer every piece of reasonings so that corresponding partial credit could be gained. Part I. True or False. Either prove the assertion or disprove it by a counter-example (6 point each): (1) The only subgroup of order 5 of a group G must be normal. (2) Every group of order 4 is cyclic. (3) A cyclic group of order 22 contains exactly 21 genarators. (4) The map A |-> (A^t)^-1 is an isomorphism of GLn(R). (5) If φ: G -> G' is a group homomorphism with |G| = 18, |G'| = 15, then |ker(φ)| equals 6 or 18. Part II. For each of the following problems, give according a short proof or an example (8 point each): (1) The functions f = 1/x, g = (x-1)/x generate a group of functions, the law of composition being composition of functions, that is isomorphic to the symmetric group S3. (2) 　　　　 ┌ 1 1 ┐　 ┌ 1 0 ┐ Matrices └ 0 1 ┘ and └ 1 1 ┘ are conjugate in GL2(R). (3) Let G be a group that contains normal subgroups of order 3 and 5, respectively. Then G contains an element of order 15. (4) Find a group G with two elements a and b, both of order 2, such that ab is of infinite order. (5) Find a group G, a normal abelian subgroup H and an element x ∈ G such that xyx^-1 = y^-1, for every y ∈ H. Part III. Give a complete proof (10 point each): (1) Let φ: G -> G' be a surjective group homomorphism with kernel K. There is a bijective correspondence between subgroups of G' and subgroups of G that contain K. (2) Let G be a group of order 25. Show that G has at least one subgroup of order 5, and that if it contains only one subgroup of order 5, then it is a cyclic group. (3) Prove that if H is a subgroup of the centre of a group G such that G/H is a cyclic group, then G is commutative. --

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