Lang theorem for algebraic groups
= Lang theorem for algebraic groups
{c}
For a connected smooth <algebraic group> and a finite-field Frobenius $F$, the <Lang map> $g\mapsto g^{-1}F(g)$ is surjective. One proof uses the twisted action $x\cdot y=xyF(x)^{-1}$. Since $dF=0$, every orbit map has surjective differential, so each orbit is open; connectedness leaves only one orbit. The same differential calculation makes the Lang map étale, with finite fibres equal to left cosets of $G^F$.