TITLE: Exotic smooth structures on Lie groups?
QUESTION [17 upvotes]: If a topological group $G$ is also a topological manifold, it is well-known (Hilbert's 5th Probelm) that there is a unique analytic structure making it a Lie group.
However, for a compact Lie group $G$, do we know if the underlying topological manifold supports any other exotic smooth structures (necessarily not a Lie group)?
Even a more specific example: Up to diffeomorphism, we have $SO(8)=SO(7)\times S^7$. If we replace the smooth structure on $S^7$ by an exotic one, do we get an exotic smooth structure on $SO(8)$?
Thank you!

REPLY [10 votes]: According to the introduction to the following paper of Farrell and Jones, if $n>4$ and $\Sigma^n$ is any exotic homotopy sphere, then $T^n\#\Sigma^n$ is not diffeomorphic to $T^n$. So lots of higher-dimensional tori have exotic structures. Farrell and Jones cite Section 15A of Wall's book on surgery for this result, but I couldn't pull it easily out of there. In any case, here's the paper reference: 
Farrell, F. T.; Jones, L. E.
Examples of expanding endomorphisms on exotic tori. 
Invent. Math. 45 (1978), no. 2, 175–179.