Prove the following: Let <a> be a cyclic group of order n. If n and m are relatively prime, then the function f(x)=x^m is an automorphism of <a>.

Hints:

1) If G=<a> and b is an element of G, the order of b is a factor of the order of a.

2) Suppose an element a in a group has order n. Then a^t = e iff t is a multiple of n (t=nq for some integer q).