Let $G$ be a linear algebraic group over a field $K$. (Say $K=\mathbb{F}_q$ or $K=\mathbb{C}$; do not assume $K$ is algebraically closed or of characteristic $0$.) Let $H_1$, $H_2$ be algebraic subgroups of $G$. Consider the multiplication map
$\phi:H_1\times H_2\to G$.
The image of $\phi$ is a constructible set, i.e., a variety $H$ with perhaps a few varieties of lower dimension deleted from it. (This is a special case of a result of Chevalley's.)
Question: when is $H_1(K) H_2(K)$ equal to the set of $K$-points of an algebraic subvariety $H$ of $G$?H(K)$? There are two issues here: closure (i.e., really getting a variety rather than a constructible set as the image) and rationality. Getting more specific, since the question above may be too hairy in general: (a) Assume that$G$is solvable. Does that help? Can we then answer the question in the affirmative? (b) Say, furthermore, that both$H_1$and$H_2$are in the same unipotent subgroup of$G$, or that$H_1$is unipotent and$H_2$is a subgroup of a corresponding maximal torus. Does that help? 2 added 2 characters in body Let$G$be a linear algebraic group over a field$K$. (Say$K=\mathbb{F}_q$or$K=\mathbb{C}$; do not assume$K$is algebraically closed or of characteristic$0$.) Let$H_1$,$H_2$be algebraic subgroups of$G$. Consider the multiplication map$\phi:H_1\times H_2\to G$. The image of$\phi$is a constructible set, i.e., a variety with perhaps a few varieties of lower dimension deleted from it. (This is a special case of a result of Chevalley's.) Question: when is$H_1(K) H_2(K)$equal to the set of$K$-points of an algebraic subgroup subvariety$H$of$G$? There are two issues here: closure (i.e., really getting a variety rather than a constructible set as the image) and rationality. Getting more specific, since the question above may be too hairy in general: (a) Assume that$G$is solvable. Does that help? Can we then answer the question in the affirmative? (b) Say, furthermore, that both$H_1$and$H_2$are in the same unipotent subgroup of$G$, or that$H_1$is unipotent and$H_2\$ is a subgroup of a corresponding maximal torus. Does that help?