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94年 - 94 淡江大學 轉學考 代數#56429
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3. Let G be a group such lhat | G | ＜ 200. Suppose G has subgroups of order 25 and 35, find the order of G.
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(B) Should Harry know whether the bottle might explode?
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【已刪除】3.The English data below provide examples of siress placement on cei.uin verbs.
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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.
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2. Prove that Z5is a lie Id.
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【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.
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5. Let G = ＜a＞ be a cyclic group of order 30，where a is a generator of G. Determine ＜a5＞ and ＜a2＞.
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6. Up to isomorphism, Jind all groups of order 4.
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7. Show that any group oi'order 35 is eyclic.
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8. Find all distinct subgroups of Z l2.
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9. Prove that every ideal of the ring of integers is principal.
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