Let p be a prime number. Prove that every power of a p-cycle is either a p-cycle or (1).
Let be a -cycle. Then you can form the subgroup (of the symmetric group) . Now . Since is a cyclic group of prime order it means the order of is either or . If it is then . Otherwise we know we can factor where 's are disjoint cycles, if is length of then . But since if (because it is prime) it means it is not possible for . Therefore and so must be a -cycle.