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ProblemSet5 - Homework 5 Due Friday Problems after the...

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Homework 5 Due: Friday 2/26/10 Problems after the double bar will not be collected but it is recommended that you do them. My assumption is that you will do them. 1. The commutator subgroup G of a group G is the subgroup generated by the set { x - 1 y - 1 xy : x, y G } . That is, every element of G has the form a i 1 1 a i 2 2 · · · a i k k where each a j has the form x - 1 y - 1 xy , each i j = ± 1 and k is any positive integer. (The commutator subgroup is the smallest normal subgroup such that G/G is abelian and thus gives a measure of how “close” G is to being abelian.) (a) Prove G G (b) Prove G/G is abelian. (c) If ϕ : G A is a homomorphism, where A is an abelian group, prove that G kerϕ . Conversely, if G kerϕ , prove that imϕ is abelian. (d) If G H G , prove that H G . 2. For n a positive integer, U ( n ) is the group consisting of the positive integers less than n and relatively prime to n with the binary operation of multiplication modulo n . For example, U (9) = ( { 1 , 2 , 4 , 5 , 7 , 8 } , · (mod 9)). Prove U (15) = Z / 4 Z × Z / 2 Z . 3. Let Q be the group of quaternions (see problem 6) and V be the Klein-4 group. Prove Q/Z ( Q ) = V . (It may help to do problem 6 first) 4. Let G
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