Let M be an o-minimal expansion of a real closed field. A definable group is a group that both the set and the graph of the operation are definable in M. Let G be a closed and bounded definable group. I will show the following:
(1) G is divisible if and only if G is definably connected.
(2) (Joint work with M.Edmundo) If G is abelian then the group structure of the torsion subgroups of G is determined.
Both proofs require the understanding of the o-minimal cohomology algebra of G.
I will also discuss the role played by the o-minimal Euler characteristic in aspects of the algebraic structure of definable groups.