# Kolchin's theorem

From Groupprops

## Statement

If a linear algebraic group is unipotent (i.e., every element is unipotent) 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 is a vector space over a field, and is a subgroup of , such that every element of is unipotent (i.e., is a nilpotent linear transformation), then one can choose a basis for in which all the element of are represented by upper triangular matrices with 1s on the diagonal.

## Related results

### Similar facts

- Borel fixed-point theorem
- Lie-Kolchin theorem: Analogous statement for solvable groups.

### Analogues for Lie algebras

- Lie's theorem: Analogous statement for solvable Lie algebras.
- Engel's theorem: Analogous statement for nilpotent linear transformations in Lie algebras.
- Kostrikin's theorem