This preview shows page 1. Sign up to view the full content.
Unformatted text preview: G . Exercise 4. [Aut( Z n ) and U ( n )  Part I] Let n N , n 2. Dene 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 )....
View
Full
Document
 Spring '10
 RyanDaileda
 Algebra

Click to edit the document details