On theorems of Lie-Kolchin, Borel, and Lang

Abstract

This chapter reviews theorems of Lie-Kolchin, Borel, and Lang. It describes two topics, the first one relates to the observation that Kolchin's proof of the Lie-Kolchin theorem can be adapted to yield an extension that has the Borel subgroup conjugacy theorem and the Borel fixed point theorem as quick consequences. The second topic is concerned with the theorem that states that if G is a connected algebraic group and σ is an endomorphism such that Gσ is finite, then 1— σ: G→G is a finite morphism. In the former topic, the development is quite elementary. It is understood that M. Sweedler has also found an easy proof of the conjugacy theorem. The latter stated theorem implies Lang's theorem that 1 — σ is surjective in case σ is the Frobenius map for some rational structure on G. The author's development yields a new, especially simple, proof of this theorem in case G is affine.