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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 Spring '10
 RyanDaileda
 Algebra, Group Theory, c., Zn, Algebraic structure

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