Does every compact connected manifold carry at most one continuous group structure? In other words, let G and G’ be compact connected Lie groups. If G and G’ are homeomorphic does that imply they are isomorphic as Lie groups?
Does uniqueness follow from classification of compact connected Lie groups? Even if so, is there a more direct way of showing uniqueness?
I know that uniqueness fails for non connected case ( for example, finite groups) and it fails for non compact case ( for example there are many distinct groups structures on Euclidean space for any $ n \geq 2$).
No. Consider, for instance, the group $U(2)$. It is homeomorphic to $SU(2)\times U(1)$. However, $U(2)$ and $SU(2)\times U(1)$ are not isomorphic Lie groups.