Why is every (reduced, separated, of finite type over an algebraically closed field $\bar{k}$) group variety $G/\bar{k}$ such that $G(\bar{k}) \cong GL_n(\bar{k})$ as groups isomorphic to $GL_n/\bar{k}$?
My idea was to use that the closed points $G(\bar{k}) \subseteq G$ are dense.