TITLE: Conditions for a smooth manifold to admit the structure of a Lie group
QUESTION [21 upvotes]: As we know, Lie group is a special smooth manifold.  I want to find some  geometric property, which is only satisfied by the Lie group. I only found one property: parallelizability.  Can you show me something more?

REPLY [4 votes]: The cohomology of the manifold has to be a Hopf Algebra.  More generally, the manifold has to be an H-space. This implies the unit sphere bundle is fibrewise homotopy-trivial (see comments below). Earlier I thought manifold + h-space implied triviality of the tangent bundle but that's not clear.   
A manifold that's an H-space is called an "H-manifold" in the literature.  Sometimes this is ambiguous since if H is a group, "H-manifold" could mean "manifold with an action of the group H" so beware when searching the literature. 
I believe there is a well-known obstruction theory for H-manifolds to be Lie groups and many algebraic topologists know a pile of examples off the top of their heads.  Unfortunately, if I ever did know such a pile of examples I've forgotten!