Past exam of the mathematics course of the University of Cambridge / 2006 / ia / Paper 3 / 2D / Solution

Both subgroups contain the identity element, so $H\cap K$ is nonempty. If $x,y\in H\cap K$, then $x{y}^{-1}$ belongs to both $H$ and $K$, hence to their intersection. This gives closure under inverses by setting $x=e$, and closure under products by using the inverse of $y$. Therefore $H\cap K$ is a subgroup of $G$, and also of both $H$ and $K$.

Lagrange's theorem for finite groups states that the order of a subgroup divides the order of its finite ambient group; more precisely $|L|=[L:M]|M|$ for $M\le L$. Applying it twice givesSince $p$ and $q$ are coprime, their only positive common divisor is one. Thus