Lang Theorem for Connected Algebraic Groups over Finite Fields
Let \(G\) be a connected algebraic group over a finite field \(F_q\), and let \(\sigma:G\to G\) be the \(q\)-Frobenius endomorphism. The Lang map \[ L_\sigma:G\to G,\qquad x\mapsto \sigma(x)x^{-1} \] is surjective.
Equivalently, every element of \(G\) is \(\sigma\)-conjugate to the identity. In Galois-cohomological language, this gives \[ H^1(F_q,G)=1 \] for connected algebraic groups.
Proof
The differential of the Frobenius morphism is zero, so the differential of \(L_\sigma\) at the identity is the negative identity after identifying tangent spaces by right translation. Translating the same computation shows that \(dL_\sigma\) is bijective at every point. Hence the image of \(L_\sigma\) contains a nonempty open subset of \(G\).
Let \(a\in G\). The same argument applied to the twisted map \[ x\mapsto \sigma(x)a x^{-1} \] shows that its image contains a nonempty open subset. Since connected algebraic groups are irreducible, the two nonempty open subsets intersect. An intersection point gives \(a=\sigma(x)x^{-1}\) for some \(x\), proving surjectivity.