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 F q {\displaystyle \mathbf {F} _{q}} , then, writing σ : G → G , x ↦ x q {\displaystyle \sigma :G\to G,\,x\mapsto x^{q}} for the Frobenius, the morphism of varieties G → G , x ↦ x − 1 σ ( x ) {\displaystyle G\to G,\,x\mapsto x^{-1}\sigma (x)} is surjective. Note that the kernel of this map (i.e., G = G ( F q ¯ ) → G ( F q ¯ ) {\displaystyle G=G({\overline {\mathbf {F} _{q}}})\to G({\overline {\mathbf {F} _{q}}})} ) is precisely G ( F q ) {\displaystyle G(\mathbf {F} _{q})} .