From Finan:
Let $\sigma\in{S_n}$, define the $\mathbf{order}$ of $\sigma$ to be the smallest positive integer such that $\sigma^m=(1)$. Prove that if $\sigma$ has order $m$ then $\tau\sigma\tau^{-1}$ has order $m$ for all $\tau\in{S_n}$.
So $\tau$ can have its own order, so let $order(\tau)=p$. Thus $\tau^p=(1)$. If $(m,p)=1$, then $order(\tau\sigma)=mp$. But doesn't $order(\tau^{-1})=p$? Thus $order(\tau\sigma\tau^{-1})=pmp$. So them this implies $(\tau\sigma\tau^{-1})^{pmp}$... Not sure where to go with this.