The groups $SO(3)$ and $SU(2)$ are usually "the" classic example of nonisomorphic Lie groups whose corresponding Lie algebras are isomorphic. While I understand the isomorphism of their algebras, it was not obvious at all to me that we must have $SU(2) \not\cong SO(3)$, I think one can conclude that by looking at the topological structure, for example the fact that their fundamental groups are nonisomorphic should suffice (I think).
What if we forget about the topology and consider them just as abstract groups? I've been trying to find some obstruction but all the facts I know about them, for example $SO(3) \cong SU(2)/Z(SU(2))$, don't seem to be enough! Am I missing something obvious?