For which positive integers $n$ can $\mathbb R P^n$ be given the structure of a topological group?
I believe that $\mathbb R P^n$ cannot be given a Lie group structure for even $n$, since then it is not orientable. But, this doesn't necessarily imply it doesn't have a topological group structure (which is not smooth); moreover, it tells us nothing about odd $n$. And ideas?