課程名稱︰代數導論二
課程性質︰系定必修
課程教師︰莊正良 教授
開課系所︰數學系
考試時間︰2005/06/27 10:00-13:00
試題：
3 2
1.(15%) How many non-isomorphic abelian groups of order 1400＝2 × 5 × 7
are there?
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2.(15%) Let A, B, C be finite abelian groups. If A⊕B＝A⊕C, show that
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B＝C. If A is infinite, give a counter-example showing that this is
false.
3.(15%) Let Ｚ be the ring of intergers. Let G be the group of units (
invertible elements with respect to multiplication) of Ｚ modulo 60.
Decompose G as a direct product of cyclic groups of prime power orders.
4.(15%) Let G be a finite group p-group. Prove that the center Z(G) of G
is nontrivial, that is, Z(G)≠{e}.
5.(15%) Let G be a finite group of order＞1 and p the minimal prime number
dividing the order of G. If H is a subgroup of G with index p, show that
H is normal in G.
6.(15%) Let G be a nonabelian group of order 105.
(a) Show that G has a normal subgroup H of order 35.
Hint: You may use Problem 4.
(b) Show that G can be presented with two generators a, b satisfying
35 3 -1 k
a ＝e, b ＝e and bab ＝a . Find all such k's.
7.(15%) Let G be a finite group and p a prime dividing |G|. Show that the
number of solutions of x^p＝e in G is a multiple of p.
Hint: If the solutions commute with each other then they form a subgroup
but in general they don't. Imitate the proof of Sylow's Theorems.
8.(15%) What is order of the group of symmetries of a regular dodecahedron?
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