Zariski density reference request

I have seen a theorem attributed to Chevalley, to the effect that a sub-semi-group in an algebraic group is Zariski dense if and only if the subgroup it generates is Zariski dense. Would anyone happen to have a reference?

• Presumably the algebraic group $G$ is over an algebraically closed field $k$ and irreducible (to avoid silliness), so it is finite type (not just locally so). The Zariski-closure of a sub-semigroup is a sub-semigroup, so you want that if $M$ is a Zariski-closed subsemigroup and $M(k)$ generates $G(k)$ then $M=G$. It suffices to show $\dim M = \dim G$. By irreducibility of $G$, it suffices to show every element of $G(k)$ is a "word" in a bounded number of letters from $M(k)$. Now see Prop. 2.2 in Ch. I of Borel's textbook on algebraic groups. Chevalley's constructible image theorem is the crux. Aug 7, 2016 at 1:44

• You can even avoid Ax-Grothendieck by observing that the translation (being an automorphism of $G$ as a variety) induces an isomorphism of $M$ with a closed subset of $G$ (hence of $M$). In other words, it is a closed immersion of $M$ into itself! The rest is an exercise (especially since $M$ is reduced) via dimension arguments. Aug 8, 2016 at 15:05