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94年 - 94 淡江大學 轉學考 代數#56429

科目：轉學考-代數 | 年份：94年 | 選擇題數：0 | 申論題數：10

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申論題 (10)

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.
2. Prove that Z5is a lie Id.
3. Let G be a group such lhat | G | ＜ 200. Suppose G has subgroups of order 25 and 35, find the order of G.
【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.
5. Let G = ＜a＞ be a cyclic group of order 30，where a is a generator of G. Determine ＜a5＞ and ＜a2＞.
6. Up to isomorphism, Jind all groups of order 4.
7. Show that any group oi'order 35 is eyclic.
8. Find all distinct subgroups of Z l2.
9. Prove that every ideal of the ring of integers is principal.
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.