Unformatted text preview: of cardinality q for each positive divisor of 12. The sizes of subgroups are 1;2;3;4;6 , and 12 . The canonical generators of these subgroups are Œ0Ł;Œ6Ł;Œ4Ł;Œ3Ł;Œ2Ł , and Œ1Ł , respectively. The inclusion relations among the subgroups of Z 12 is pictured in Figure 2.2.2 on the next page . Corollary 2.2.28. Let b 2 Z , b ¤ . (a) The cyclic subgroup h ŒbŁ i of Z n generated by ŒbŁ is equal to the cyclic subgroup generated by ŒdŁ , where d D g : c : d :.b;n/ . (b) The order of ŒbŁ in Z n is n= g : c : d :.b;n/ . (c) In particular, h ŒbŁ i D Z n if, and only if, b is relatively prime to n . Proof. One characterization of d D g : c : d :.b;n/ is as the smallest positive integer in f ˇb C ±n W ˇ;± 2 Z g . But then d is also the smallest of positive...
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- Fall '08
- Algebra, Prime number, Cyclic group, OED, 0 j