課程名稱︰代數導論一
課程性質︰數學系必修
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︰2011年10月20日
考試時限：180 分鐘
是否需發放獎勵金：是
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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