About: Lang's theorem

In algebraic geometry, Lang's theorem, introduced by Serge Lang, states: if G is a connected smooth algebraic group over a finite field , then, writing for the Frobenius, the morphism of varieties is surjective. Note that the kernel of this map (i.e., ) is precisely . The theorem implies that vanishes, and, consequently, any G-bundle on is isomorphic to the trivial one. Also, the theorem plays a basic role in the theory of finite groups of Lie type. The proof (given below) actually goes through for any that induces a nilpotent operator on the Lie algebra of G.