Lang–Steinberg theorem

In algebraic geometry, Lang's theorem, introduced by Serge Lang, states: if G is a connected smooth algebraic group over a finite field ${\displaystyle \mathbf {F} _{q}}$, then, writing ${\displaystyle \sigma :G\to G,\,x\mapsto x^{q}}$ for the Frobenius, the morphism of varieties

is surjective. Note that the kernel of this map (i.e., ${\displaystyle G=G({\overline {\mathbf {F} _{q}}})\to G({\overline {\mathbf {F} _{q}}})}$) is precisely ${\displaystyle G(\mathbf {F} _{q})}$.

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