1. ## cycle

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

$\displaystyle wlog$, 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

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

$\displaystyle wlog$, 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
I consider this proof to be fine. You can add by showing the the LHS and the RHS agree for all values of $\displaystyle 1,2,...,m$ in other words evaluating the LHS for any $\displaystyle 1,2,...,m$ will give the same result as RHS and so the two functions (bijections) are the same.