In particular, I am wondering if $H^1_{DR}(G)\neq 0$ implies that the group can written as a semidirect product of $\mathbb{S^1}$ and something else, with the $\mathbb{S^1}$ factor being responsible for the first cohomology group. I have no idea if this is true. In case it is not clear by $H^1_{DR}(G)$, I mean the De Rham cohomology of the underlying manifold associated to a group $G$. (I understand this is isomorphic to some kind of Lie algebra cohomology, but as of now know nothing about Lie algebra cohomology).

Note that I originally asked this on mathstackexchange and received no answers so I hope this was because perhaps it was more appropriate for here... I have deleted the question on stackexchange.