I am working on ex. 4.2.11 in Dummit and Foote's Abstract Algebra book. The question reads:
Let $G$ be a finite group and let $\pi : G\rightarrow S_{G}$ be the left regular representation. Prove that if $x$ is an element of $G$ of order $n$ and |$G$|=$mn$, then $\pi (x)$ is a product of $m$ $n$-cycles. Deduce that $\pi (x)$ is an odd permutation if and only if |$x$| is even and $\frac{|G|}{|x|}$ is odd.