TITLE: What's the sufficient or necessary conditions for a manifold to have Lie group structure?
QUESTION [9 upvotes]: For example, given a Lie group, its fundamental group must be Abelian. So $\Sigma_g$ ($g>1$)  can't have Lie group structure. We also know for  $S^n$ only $n=0,1,3$ can have Lie group  structures. 
In general, what's the sufficient or necessary conditions for a manifold to have Lie group structure?

REPLY [10 votes]: Compact Lie groups are finite loop spaces, but Andersen, Bauer, Grodal, Pedersen (Invent. Math., 2004) give an example of a finite loop space that is not (even rationally) homotopy equivalent to any compact Lie group.