Show that if $H$ and $K$ are subgroups of a group $G$, then $H \cap K$ is also a subgroup of $G$. Show also that if $H$ and $K$ have orders $p$ and $q$ respectively, where $p$ and $q$ are coprime, then $H \cap K$ contains only the identity element of $G$. [You may use Lagrange's theorem provided it is clearly stated.]