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）︰2011/10/20 考試時限（分鐘）：180 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 examination, G denote a group, H, K denote subgroups of G, and e always is the identity of G. (1) (30 points) "yes" or "no". Either give a brief reason or give a counter example. 5 points each. (a) If a^2=e for every a∈G, then G is commutative (abelian). (b) If G is commutative and there are a, b∈G with o(a) = m, o(b) = n, then there exists some c∈G with o(a) = [m,n], the least common multiple of m and n. (c) Every group of order 4 is commutative. (d) Every group of order 6 is commutative. (e) Every group of order 8 is commutative. (f) If both H and K are normal subgroups, then so are H‧K and H∩K. (2) (15 points) Suppose whenever Ha≠Hb then aH≠bH. Prove that H is a normal subgroup. (3) (15 points) Suppose H and K are of finite index in G. Show that H∩K is also finite index in G. (4) (15 points) Let G be a finite abelian group of order relatively prime to a natural number n. Prove that every g∈G can be written as g=x^n for a unique x∈G. (5) (15 points) Prove that a group of order 9 is abelian. (6) (10 points) (a) Find a prime number p such that p ┤f(k) for every inter k. (b) Prove that for each prime number p, there exists some integer k such that p│g(k). --

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