I wish to thank Professor Claudio Gorodski for his very helpful answers to my question on the webcite: If compact connected Lie groups are homeomorphic as topological space, are they isomorphic as Lie groups?

He said: Let $G_{1}$ and $G_{2}$ be two compact, connected Lie groups with isomorphic homotopy groups in each dimension. Then their Lie algebras are isomorphic.

Now my question is: If $G_{1}$ and $G_{2}$ are two compact, connected topological groups which are homeomorphic as topological space, are there any isomorphism theorems?