Let $K$ be a field of characteristic zero. Let $G/K$ be a group scheme of finite type. Assume that $G$ is commutative and connected. For a natural number $n$ denote by $n_G: G\to G$ the multiplication by $n$ morphism. Is it true that $n_G$ is surjective with finite kernel?
(I know that the answer is yes provided $G$ is an abelian variety. But the proof of this fact makes use of a very ample invertible sheaf on $G$, so it does not carry over to the general case directly.)