left regular representation(group action) of a finite group $G$ Dummit Foote 4.2.11 Let $G$ be a finite group and let $\pi : G\rightarrow S_G$ be the left regular representation.
Question is to :
Prove that if $x$ is an element of order $n$ and $|G|=mn$ then $\pi(x)$ is a product of $m$ cycles of length $n$.Deduce that $\pi(x)$ is an odd permutation iff $|x|$ is even and $\frac{|G|}{|x|}$ is odd.
I do not have much to say about what i have tried on the way to solution.
what i am thinking is to consider $H=\big<x\big>$ and consider restricted action on cosets of $H$ in $G$.
I am not going anywhere in this way.
please help ,e by givinig some hint for this.
i would be thankful if some one can make me writeup the solution (by giving some hints) than just write an answer in the answer block.
Thank You
 A: Let's break the question into two parts. First you need to show that if $x$ has order $n$, then $\pi(x)$ is a product of $m$ cycles of length $n$. So how does $x$ act on $G$? If you pick $g \in G$, then we get the elements $g, xg, x^2g, ..., x^{n-1}g$. Show that these are all distinct since otherwise the order of $x$ is smaller than $n$. Thus, the action of $x$ splits $G$ into $m$ sets of $n$ elements. These are nothing more than the cosets of $\langle x \rangle$, and these cosets are your $n$-cycles. In particular, $(g, xg, x^2g, ..., x^{n-1}g)$ is an $n$-cycle.
Second part. Show that $\pi(x)$ is an odd permutation if and only if $|x|$ is even and $\frac{|G|}{|x|}$ is odd. So in terms of $n$ and $m$ this means $n$ is even and $m$ is odd. Ok, $\pi(x)$ is the product of $m$ cycles of length $n$. Each cycle of length $n$ gives a permutation of $G$ (where you fix everything not in the cycle). We have $sgn(\sigma_1 \circ \sigma_2)=sgn(\sigma_1)\cdot sgn(\sigma_2)$ for permutations $\sigma_1, \sigma_2$. Thus, the sign of $\pi(x)$ is the product of the signs of the $n$-cycles.  Show that an $n$-cycle has sign $-(-1)^n$. So the sign of $\pi(x)$ is just $[-(-1)^n]^m$. Show this is $-1$ precisely when $m$ is odd and $n$ is even.
