Let $G$ be an algebraic group defined over some algebraically closed field $\mathbb{K}$ and also over some subfield $k\subset \mathbb{K}$. There is thus a natural group structure on the set of $k$-points of $G$. It seems impossible to me that this group is cyclic and infinite. Is this true?

Let $G$ be a linear algebraic group defined over some algebraically closed field $\mathbb{K}$ and also over some subfield $k\subset \mathbb{K}$. There is thus a natural group structure on the set of $k$-points of $G$. It seems impossible to me that this group is cyclic and infinite. Is this true?

EDIT: user48841 explained that this happens with elliptic curves. I am in fact interested in linear algebraic groups.