StudyGuideDirectProducts

# StudyGuideDirectProducts - and n are relatively prime then...

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Direct Products Study Guide Outline 1. External Direct Products If G and H are groups, the direct product of G and H , denoted G × H (or G H in the book), is defined as follows: The elements of G × H are ordered pairs ( g, h ), where g G and h H . The operation on G × H is defined componentwise, i.e. ( g 1 , h 1 )( g 2 , h 2 ) = ( g 1 g 2 , h 1 h 2 ) . The order of G × H is the product of the orders of G and H : | G × H | = | G | | H | Problems: 10, 11, 12 2. Orders of Elements Let G and H be groups, and let ( g, h ) G × H . Then the order of ( g, h ) is the least common multiple of the orders of g and h : ( g, h ) = lcm ( | g | , | h | ) Problems: 13, 14, 15, 16 3. Internal Direct Products Let G be a group, let H and K be subgroups of G , and suppose that 1. H K = { e } , 2. Every element of G can be expressed as the product of an element of H with an element of K , and 3. Elements of H commute with elements of K . Then G H × K . This theorem lets us recognize that a given group is a direct product. For example, V Z 2 × Z 2 and D 6 D 3 × Z 2 More generally, D 2 n D n × Z 2 for odd values of n .

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4. Direct Products of Cyclic Groups
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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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