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Jérémy Blanc
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Non existence of cyclic infinite linear algebraic groups

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.

Jérémy Blanc
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