I have to show that if $\displaystyle sigma$ is a cycle of odd length then $\displaystyle sigma^2$ is a cycle.

So this is what I assume, (and please help me check if it is an suitable answer);

$\displaystyle wlog$, assume that

$\displaystyle sigma=(1,2,3,......,m) where m is odd.$

Because m is odd, I can compute that

$\displaystyle sigma^2=(1,2,3,....,m)(1,2,3,....,m)(1,3,5,....,m, 2,4,6,...m-1) $

which is again a cycle.

Is this a suitable proof, and what else can I add to it.

Thank You