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Unformatted text preview: 208 3. PRODUCTS OF GROUPS Remark 3.6.27. Note that while the proof insures that the group of units of K is cyclic, it does not provide a means of actually finding a generator! In particular, it is not obvious how to find a nonzero element of Z p of multiplicative order p 1 . Proposition 3.6.28. (a) If N has prime decomposition N D p k 1 1 p k 2 2 p k s s , then .N/ .p k 1 1 / .p k 2 2 / .p k s s /: (b) .2/ and .4/ are cyclic. .2 n / Z 2 Z 2 n 2 if n 3 . (c) If p is an odd prime, then for all n , .p n / Z p n 1 .p 1/ Z p n 1 Z p 1 . Proof. Part (a) follows from Example 3.1.4 and induction on s . The groups .2/ and .4/ are of orders 1 and 2, respectively, so they are necessarily cyclic. For n 3 , we have already seen in Example 2.2.34 that .2 n / is not cyclic and that .2 n / contains three distinct elements of order 2, and in Exercise 2.2.30 that OE3 has order 2 n 1 in .2 n / . The cyclic subgroup h OE3 i contains exactly one of the three elements of order 2. Ifcontains exactly one of the three elements of order 2....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08