Let $G$ be a group and $U,H\leq G$ subgroups, such that $H$ is of finite Index in $G$ (not necessary U, too). May $n=|G:H|$. One can easily show with an Injection between the two appropriate cosets that $|U:H\cap U|\leq n$, but does $|U:H\cap U|\mid n$ hold? So does the Index of the Intersection $H\cap U$ in $U$ divides the Index of $H$ in $G$?
I don't think that the claim holds, so I'm searching for an example to disprove it. Is there a difference between the two cases, where $G$ is infinite or finite? Thanks for help.