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 aba1 b1 ∈ 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 : aHa1 = 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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 Spring '10
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 Algebra, Group Theory, Addition, Normal subgroup, Abelian group, Cyclic group, automorphism

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