If $H\leq G$ of index $p$ in $G$ and $p\mid |G|$ then $H\trianglelefteq G$

Question

Given that $H\leq G$ of index $p$ in $G$, where $p$ is the smallest prime integer such that $p \mid |G|$, then $H \trianglelefteq G$. I would appreciate some hints, as I don't even know where to begin. We can use a group action of $G$ on the left cosets of $H$ by left multiplication, but I don't know how to move further, and, particularly, how to use the fact that $p$ is the smallest prime dividing the order of $G$.