Kolchin's theorem

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 $V$ is a vector space over a field, and $G$ is a subgroup of $GL(V)$, such that every element $g$ of $G$ is unipotent (i.e., $g-1$ is a nilpotent linear transformation), then one can choose a basis for $V$ in which all the element of $G$ 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