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# hw10 - G Exercise 4[Aut Z n and U n Part I Let n ∈ N n...

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Modern Algebra 1 Homework 5.1 Spring 2010 Due February 17 Exercise 1. Let f : G H be an isomorphism of groups. a. Let a G . Prove that | a | = | f ( a ) | . b. Prove that G is abelian if and only if H is abelian. Exercise 2. Let f : G H be an isomorphism of groups. Prove that the mapping K 7→ f ( K ) gives a bijection between the sets of subgroups of G and H , and that K J G if and only if f ( K ) f ( J ) H . Exercise 3. Let G be a group and let a, b G . The commutator of a and b is the group element [ a, b ] = aba - 1 b - 1 . Let f : G H be a group homomorphism. Prove that Im f is abelian if and only if ker f contains every commutator in
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Unformatted text preview: G . Exercise 4. [Aut( Z n ) and U ( n ) - Part I] Let n ∈ N , n ≥ 2. Deﬁne U ( n ) = { k ∈ Z n | gcd( n,k ) = 1 } . Let f ∈ Aut( Z n ). a. Show that Z n = h k i if and only if gcd( n,k ) = 1. [ Hint: Such a k must have | k | = n , and we proved a theorem giving the order of any element in Z n .] b. Prove that for all k ∈ Z n we have f ( k ) = kf (1). Conclude that Z n = h f (1) i . c. Conclude that f (1) ∈ U ( n )....
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