ProblemSet6 - G 4 We stated in class that if G H and K are...

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Homework 6 Due: Monday 3/15/10 1. Let G = { 1 , 7 , 43 , 49 , 51 , 57 , 93 , 99 , 101 , 107 , 143 , 149 , 151 , 157 , 193 , 199 } under multiplication modulo 200. Express G as a direct product of cyclic groups. 2. Suppose G is an Abelian group of order 9. What is the maximum number of elements (excluding the identity) one needs to compute the order of to determine the isomorphism class of G ? What if G has order 18? 3. Suppose G is an Abelian group of order 16 and in computing the orders of its elements you come across an element of order 8 and two elements of order 2. Explain why no further computations are needed to determine the isomorphism class of
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Unformatted text preview: G . 4. We stated in class that if G , H and K are nite groups and G H = G K then H = K . Show that if we allow G to be innite then H need not be isomorphic to K . 5. Dirichlets theorem say that, for every pair of relatively prime integers a and b , there are innitely many primes of the form at + b . Use Dirichlets theorem to prove that every nite Abelian group is isomorphic to a subgroup of U ( n ) for some n . (Note: If m and n are relatively prime then U ( mn ) = U ( m ) U ( n ).)...
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