College Algebra Exam Review 198

College Algebra Exam Review 198 - 208 3. PRODUCTS OF GROUPS...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

Ask a homework question - tutors are online