It's hard...

Oh, you must know group theory and symmetry groups in order to attempt this.

Let sigma = (1 2 3 ... n) be in S_n (a cycle) with n>=1. Let k be a positive integer.

Show that if k is relatively prime to n, then sigma^k (with the operation being composition, obviously) is a cycle, that is, sigma applied k times can be written as one cycle of length greater than one composed with possibly a bunch of disjoint cycles of length 1.

Any ideas?