exam1 - a homomorphism(b Suppose that G is a group and x 2...

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MATH 4107 EXAM #1 February 19, 2010 PLEASE READ THESE DIRECTIONS: Answer PROBLEM 1 (20 points) and choose TWO other problems to answer (15 points each). You may also answer (for up to 5 points extra credit) ONE additional problem. In this case, please specify which problem is the extra credit problem. All statements require proof or justi±cation. There are 50 points total, plus up to 5 points of extra credit. 1. The two parts of this problem are not related. (a) Let G = { 2 n : n Z } . This is a group with respect to the operation of ordinary multiplication of numbers (you do not need to prove that). Prove that G is isomorphic to the group of integers Z (the operation on Z is addition of numbers). Note: When you de±ne your isomorphism, you can just state that it is a bijection. You don’t have to prove the bijectiveness (as long as it really is a bijection!), just prove that it is
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Unformatted text preview: a homomorphism. (b) Suppose that G is a group and x 2 = e for every x ∈ G . Prove that G must be abelian. 2. Suppose that H is a subgroup of a ±nite group G , and the index of H in G is i G ( H ) = 2. (a) Show that if g 2 ∈ gH then g ∈ H. (b) Show that g 2 ∈ H for every g ∈ G . Hint: There are two cases to consider: g ∈ H and g / ∈ H. 3. Suppose N is a normal subgroup of a group G and we know that aba-1 b-1 ∈ N for every a, b ∈ G. Show that G/N is abelian. Hint: Don’t try to show that G is abelian! You want to show that the quotient group G/N is abelian, i.e., ( Na )( Nb ) = ( Nb )( Na ) for every a , b ∈ G. 4. Let H be a subgroup of a group G, and de±ne N = { a ∈ G : aHa-1 = H } . Prove that: (a) N is a subgroup of G, (b) H ⊆ N, and (c) H is a normal subgroup of N. 1...
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