Suppose G is an finite group. Let N be a normal subgroup of G and A be an arbitraty subgroup. Prove that $$\mid AN\mid=\frac{\mid A\mid\mid N\mid}{\mid A\cap N\mid}$$  My initial approach is that I should somehow invoke one of the isomorphism theorems that involves quotient group and then apply another theorem to grasp about its size. Yet I've no idea how to actually approach it, especially with the given condition that $N$ is normal.
Let $H$, $K$ be subgroups of $G$. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.  I need this theorem to prove something.
Let $G = \mathbb{R} / \mathbb{Q}$. Is this an interesting group to study?  Is it isomorphic to any more natural mathematical objects?