The question is from Grillet's lovely Abstract Algebra:
Let $A,B$ be two cosets of possibly different normal subgroups of finite index of a group $G$. It is to be shown that $A \cap B$ is either empty or the coset of a normal subgroup of $G$, also of finite index.
So far I have assumed that the intersection is not empty and that the normal subgroups are different. In this case, we can write:
$xN \cap yH = \{ z: \exists n \in N, h \in H : z=xn=yh \}$
where $N,H$ are our normal subgroups of finite index and $x,y \in G$ fixed. I haven't really gotten further. I've also tried applying the second isomorphism theorem but it doesn't seem to stick. A hint or some guidance would be welcome.