Two related elementary facts in group theory are sometimes called Poincaré's theorems.
- If $H\lneq G$ and $[G:H]<\infty$, then there is $N\leq H$, $N\lhd G$ such that $[G:N]<\infty$.
- The intersection of a finite number of subgroups of finite index is of finite index.
Did he prove both? Could you please give me references to the paper(s) (or at least the year(s))? I'm especially interested in the first one, but I would prefer to know the whole story.