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Lang–Steinberg theorem

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In mathematics, the Lang–Steinberg theorem, introduced by Lang (1956) for the special case of the Frobenius endomorphism F and by Steinberg (1968) in general, gives conditions for the Lang map g → g−1F(g) of an endomorphism F of an algebraic group to be surjective.

Statement

Suppose that F is an endomorphism of an algebraic group G. The Lang map is the map from G to G taking g to g−1F(g).

The Lang–Steinberg theorem states that if F is surjective and has a finite number of fixed points, and G is a connected affine algebraic group over an algebraically closed field, then the Lang map is surjective.

See also

References

  • Lang, Serge (1956), "Algebraic groups over finite fields", American Journal of Mathematics, 78: 555–563, ISSN 0002-9327, MR0086367
  • Steinberg, Robert (1968), Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, Providence, R.I.: American Mathematical Society, MR0230728
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