Define a permutation $\alpha\in S_n $ to be regular if either $\alpha$ has no fixed points and it is the product of disjoint cycles of the same length, or $\alpha=(1)$.
Prove that $\alpha$ is regular if and only if $\alpha$ is a power of an n-cycle.
It is a homework question in J.J.Rotman's book: A first course in abstract algebra with applications.
I have just begun reading the book, and I find this question confusing----I've tried using induction, but couldn't figure out a good way. I would really appreciate your help.