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94年 - 94 淡江大學 轉學考 代數#56429
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5. Let G = ＜a＞ be a cyclic group of order 30，where a is a generator of G. Determine ＜a
5
＞ and ＜a
2
＞.
其他申論題
Show all your work. 10 points each.1 • Show that a group G is abelian if and only if x2 = e for any x in G wlieie e is Ihe identity of G.
#214332
2. Prove that Z5is a lie Id.
#214333
3. Let G be a group such lhat | G | ＜ 200. Suppose G has subgroups of order 25 and 35, find the order of G.
#214334
【已刪除】4. Let G be the set of ail rational numbers except -1. Show that (G, ) is a group, where ab = a + b + ab for all a and b in G.
#214335
6. Up to isomorphism, Jind all groups of order 4.
#214337
7. Show that any group oi'order 35 is eyclic.
#214338
8. Find all distinct subgroups of Z l2.
#214339
9. Prove that every ideal of the ring of integers is principal.
#214340
10.Let G be a group and a and b are h G. Suppose a2 = e and ab4 a = b7. Show that b33 = e, where e is tlie identity of G.
#214341
（一）10110111 是代表 10 進位多少的整數？【3 分】
#214342
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