Group Theory – Prove Group of Order 2m Contains Normal Subgroup of Order m

Question

I searched in the existing post and didn't find this problem. I am sorry if someone else have already posted.

Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$.

There is a hint:

Denote by $\rho$ the regular represetation of $G$: find an odd permutation in ${\rho}(G)$.

I don't know how to find an odd permutation in the regular representation. I am wondering whether all the elements of $G$ of odd order form this subgroup in this case.

Thanks.

Answer 1

By Cauchy's theorem, $G$ contains an element of order 2. How does this act in the regular representation?