I know that the fundamental group of orientable surfaces is of the form $F(a_1,b_1,\ldots,a_n,b_n)/N(\prod a_j*b_j*a_j^{-1}*b_j^{-1})$ where $F$ is the free products with certain generators and $N$ is the normalizer of certain group.
Similarly, the fundamental group of non-orientable surfaces is $F(a_1,\ldots,a_n)/N(\prod a_j^2)$.
My understanding of these groups is not very precise yet. Why is it the case that only the sphere and the projective space which have fundamental group $0$ and $\mathbb{Z}_2$ the only finite groups in this list?