課程名稱︰ 代數導論一
課程性質︰ 系定必修
課程教師︰ 黃漢水教授
開課學院： 理學院
開課系所︰ 數學系
考試時間︰ 2006/11/16 12:20-15:10
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
一、Suppose that α,β belong to S5 such that
α = (1 2 3 4 5), β = (1 3)(4 5).
Let H = <α>, K = <β>.
(1) Prove or disprove that there is a positive integer k such that
βα = (α^k)β (10%)
(2) Prove or disprove that HK = { ab | a belongs to H, b belongs to K}
is a subgroup of S5. (10%)
二、A group G has subgroups of order 8 and 20 and |G| ≦ 60. What can you
conclude about |G|？ (15%)
三、Suppose that G is a group, H,K are subgroups of G and a,b belong to G.
Prove or disprove that if aH = bK then H = K. (20%)
四、Suppose that H is a subgroup of a group G such that (G:H) = 2. Prove that
if a doesn't belong to H and b doesn't belong to H, then ab belongs to H.
(15%)
五、(1) Let G be a group and an isomorphism of G onto G is called an
automorphism. Prove that the set Aut(G) of all automorphisms of G is
itself a group with respect to composition. (10%)
(2) Let G = <α> be a cyclic group with generator α of order m.
Then prove that Aut(G) is an abelian group. (10%)
(3) Let G = <α> be a cyclic group with generator α of order 17.
Then what is the group Aut(G). (10%)
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讓我們擬設一些想像：
同樣星映之下處女座的我卻迷戀虛空，
迷上的月涼清語，迷上親吻繁星。
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