Let $k$ be a finite field and $G$ a connected, reductive linear algebraic group defined over $k$. It is well-known that the union of the maximal tori of $G$ is dense in $G$ (more generally, if $G$ is not reductive, the union of its Cartan subgroups is dense).

It is also true, that $G$ is generated by the maximal tori defined over $k$ (this is proven in a paper by Borel and Springer from 1968).

However, is the union of the maximal tori which are defined over $k$ still dense in $G$?

`$k$`

(so their union is closed and of dimension equal to the rank of`$G$`

): in concrete terms, such a torus is the zero set in`$G$`

of finitely many polynomials with coefficients in`$k$`

. For more detailed discussion see for instance the Springer-Steinberg article (section II.1) in the 1970 Lect. Notes in Math. 131 (Springer). $\endgroup$ – Jim Humphreys May 20 '11 at 17:25