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Unformatted text preview: and n are relatively prime, then Z mn ≈ Z m × Z n This makes it possible to write any cyclic group as a direct product of cyclic groups whose orders are prime powers. For example, Z 360 ≈ Z 8 × Z 9 × Z 5 . This is most often used to check whether two direct products of cyclic groups are isomorphic. Problems: 21, 22 5. Structure of Ugroups If m and n are relatively prime, then U ( mn ) ≈ U ( m ) × U ( n ) This makes it possible to express any Ugroup as a direct products of Ugroups for prime powers. For example, U (360) ≈ U (8) × U (9) × U (5). If p is prime, then U ( p ) ≈ Z p1 More generally, if p is an odd prime, then U ( p k ) ≈ Z φ ( p k ) where φ ( n ) =  U ( n )  . Things work slightly diﬀerently for powers of two: U (2 k ) ≈ Z 2 × Z 2 k2 Problems: 17, 18, 19, 20...
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 Spring '09
 Algebra, Prime number, Cyclic group, Cyclic Groups, direct product, Direct Products

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