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/ Š ˚.p k 1 1 / ˚.p k 2 2 / ˚.p k s s /: (b) ˚.2/ and ˚.4/ are cyclic. ˚.2 n / Š Z 2 Z 2 n 2 if n 3 . (c) If p is an odd prime, then for all n , ˚.p n / Š Z p n 1 .p 1/ Š 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 OE3Ł has order 2 n 1 in ˚.2 n / . The cyclic subgroup h OE3Ł 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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- Fall '08
- Algebra, Prime number, n1, Cyclic group, hŒ3i