Lang theorem for algebraic groups

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 $G^F$.

For the Lang map, ${\mathcal{L}}_F(gh)=h^{-1}{\mathcal{L}}_F(g)F(h)$. This twisted right-translation identity determines which subgroups act on its inverse images. Fibres are left $G^F$-cosets, by the Lang theorem for algebraic groups.