Kolchin's theorem

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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

Analogues for Lie algebras