Unipotent matrix

A square matrix $ A $ over a ring for which the matrix $ A - I _ {n} $, where $ n $ is the order of $ A $, is nilpotent, i.e. $ ( A - I _ {n} ) ^ {n} = 0 $. A matrix over a field is unipotent if and only if its characteristic polynomial is $ ( x - 1) ^ {n} $.

A matrix group is called unipotent if every matrix in it is unipotent. Any unipotent subgroup of $ \mathop{\rm GL} ( n, F ) $, where $ F $ is a field, is conjugate in $ \mathop{\rm GL} ( n, F ) $ to some subgroup of a special triangular group (Kolchin's theorem). This assertion is also true for unipotent groups over a skew-field, if the characteristic of the latter is either 0 or greater than some $ \gamma ( n) $.