Element structure of unitriangular matrix group of degree three over a finite discrete valuation ring

This article describes the element structure of the unitriangular matrix group of degree three over a finite discrete valuation ring. It builds on the discussion at element structure of unitriangular matrix group of degree three over a finite field.

We assume that the residue field has size $$q$$ and characteristic $$p$$, with $$r = \log_pq$$. We denote by $$l$$ the length of the discrete valuation ring. The size of the ring is thus $$q^l = p^{rl}$$ and the order of the field is $$q^{3l} = p^{3rl}$$.