How to find $\sigma$ with $\tau=\sigma^k$?

Question

This question is from Dummit-Foote:

If $\tau=(1\ 2)(3\ 4)(5\ 6)(7\ 8)(9\ 10)$ determine whether there is a $n$-cycle $\sigma$ $(n\ge 10)$ with $\tau=\sigma^k$ for some integer $k$.

Is there any other method to find $\sigma$ than trial and error?

Answer 1

Experiment with powers of cycles in order to see what happens. In particular, all new cycle types of the powers are obtained be considering only powers $\rho^d$ where $d$ is a divisor of $|\rho|$.

Say $\rho=(1~2~3~4~5~6)$ within $S_6$ for instance. What are $\rho^2$ and $\rho^3$? In general, if $\rho$ is an $n$-cycle and we have a divisor $d\mid n$, can you predict the cycle type of $\rho^d$? Apply this understanding to your problem.