Question

Let G be a zariski dense subgroup of algebraic group H. Then is H the zariski closure of G? Conversely, If G is a subgroup of the algebraic group H and if H is the zariski closure of G, then is G Zariski dense in H?

What are the equivalent conditions of being zariski dense subgroup of an algebraic group?

Thanks in advance.

Answer 1

I am also not sure, whether I understand the question properly; but it seems to me that one should distinguish carefully between the group scheme and its geometric points here.

To fix ideas, let $K$ be an algebraically closed field and $G/K$ an algebraic group. An algebraic subgroup of $G$ is by definition a closed subgroup scheme of $G$. Furthermore there is the abstract group $G(K)$, and we have the Zariski topology on $G(K)$. There is a bijection $H\mapsto H(K)$ from the set of smooth algebraic subgroups of $G$ to the set of closed subgroups of $G(K)$. (Cf. Milne's lecture notes on algebraic groups for example.)

If $\Gamma$ is a subgroup of $G(K)$, then there is a unique smooth algebraic subgroup $\Gamma^{zar}$ of $G$, such that $\Gamma^{zar}(K)=\overline{\Gamma}$. Some people call the algebraic group $\Gamma^{zar}$ the Zariski closure of $\Gamma$.

Assume now that $G$ is smooth. With the above terminology we see:

a) If $\Gamma$ is Zariski dense in $G(K)$, then $\Gamma^{zar}=G$.

b) If $H$ is a smooth algebraic subgroup of $G$ and $H(K)$ is Zariski dense in $G(K)$, then $H(K)=G(K)$ and consequently $H=G$.

And one cannot drop the smoothness assumption on $G$ here.