I'm having difficulty expressing the proof, even though the statement to be proved seems obvious to me.

Prove: $\displaystyle \alpha^2$ is a cycle iff the length of the cycle $\displaystyle \alpha$ is odd.

What I've got in my head is that if $\displaystyle \alpha=(a,b)$ then $\displaystyle \alpha^2=(a)(b)$. Then if we add another 2 elements c and d, $\displaystyle \alpha=(a,b,c,d)$, making $\displaystyle \alpha^2=(a,c)(b,d)$. The pattern will obviously continue, giving 2 disjoint cycles where the elements alternate for a beginning cycle of any even length. I just need some help assembling my rambling intuitive understanding of the problem into a proper proof.

EDIT: fixed tex tags