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:<math>G \to G, \, x \mapsto x^{-1} \sigma(x)</math>
is surjective. The theorem implies that <math>H^1(\mathbf{F}_q, G) = H_{\text{ét}}^1(\operatorname{Spec}\mathbf{F}_q, G)</math>
vanishes,<ref>This is "unwinding definition". Here, <math>H^1(\mathbf{F}_q, G) = H^1(\operatorname{Gal}(\overline{\mathbf{F}_q}/\mathbf{F}_q), G(\overline{\mathbf{F}_q}))</math> is [[Galois cohomology]]; cf. Milne, Class field theory.</ref> and, consequently, any [[torsor (algebraic geometry)|''G''-bundle]] on <math>\operatorname{Spec} \mathbf{F}_q</math> is isomorphic to the trivial one. Also, the theorem plays a basic role in the theory of [[finite groups of Lie type]].
It is not necessary that ''G'' is affine. Thus, the theorem also applies to [[abelian variety|abelian varieties]] (e.g., [[elliptic curve]]s.) In fact, this application was Lang's initial motivation.
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