Lie-Kolchin theorem

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Statement

If a connected linear algebraic group over an algebraically closed field is solvable then we can choose a basis for the vector space in which the whole group becomes a subgroup of the group of upper triangular matrices.

More explicitly:

If V is a vector space over an algebraically closed field K, and G is a solvable connected closed subgroup of GL(V), then one can choose a basis for V in which all the element of G are represented by upper triangular matrices.

Related results

Similar facts

Analogues for Lie algebras

Facts used

  1. Borel-Morozov theorem which in turn uses the Borel fixed-point theorem: The theorem states that any Borel subgroup contains a conjugate of any connected solvable closed subgroup

Proof

The proof follows directly from Fact (1), applied to the group GL(V).