A finite group acting on a torsion-free group

Question

I'm trying the following problem

Let $G$ be a group. $H, K$ subgroups of $G$, where $|K|=m$, $H$ is torsion-free and $[G:H]=n$. Prove that $m\leq n$ and $m\mid n$.

The hint: Use some group action of $K$ over $G/H$.

I tried to use the action defined by left multiplication of cosets in $G/H$ by elements of $K$ (I guess it's the natural action), but I can't conclude anything useful.

Can someone suggest any idea?

Answer 1

2nd hint : Show that the natural action of $K$ on $G/H$ is free.