TITLE: Differential structures on compact Lie groups
QUESTION [5 upvotes]: Given a compact Lie group can there be a differential structure on it with respect to which one cannot define a smooth group operation?

REPLY [5 votes]: This is definitely not my expertise, but here is a stab at it.
A solution to Hilbert's 5th Problem states that given a topological group that is also a manifold there is a unique way to give it the structure of a Lie group. 
On the other hand, there are exotic structures on Lie groups (including compact ones).
So take a Lie group $G$ that admits an exotic structure.  Let $M$ denote the exotic version of $G$.  Then $G$ and $M$ are homeomorphic but not diffeomorphic.  However, if $M$ admitted a smooth group operation, then we contradict the uniqueness part of the solution to Hilbert's 5th Problem.