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 $\gamma ( n)$.