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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 U-groups If m and n are relatively prime, then U ( mn ) ≈ U ( m ) × U ( n ) This makes it possible to express any U-group as a direct products of U-groups for prime powers. For example, U (360) ≈ U (8) × U (9) × U (5). If p is prime, then U ( p ) ≈ Z p-1 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 k-2 Problems: 17, 18, 19, 20...
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This document was uploaded on 11/03/2010.
- Spring '09