Element structure of unitriangular matrix group of degree four over a finite field

This article describes in detail the element structure of the unitriangular matrix group of degree four over a finite field. We denote the field size by $$q$$, the field characteristic by $$p$$, and the value $$\log_pq$$ by $$r$$. Further, we denote the group as $$UT(4,q)$$.

Number of conjugacy classes
The general theory says that number of conjugacy classes in unitriangular matrix group of fixed degree over a finite field is polynomial function of field size, where the degree of the polynomial is one less than the degree of matrices. Thus, we expect that the number of conjugacy classes is a polynomial function of $$q$$ of degree 4 - 1 = 3. Indeed, this is the case, and the explicit polynomial is $$2q^3 + q^2 - 2q$$.

Conjugacy class structure in the unitriangular matrix group
For the right-most column for the type of matrix, we use the (row number, column number) notation for matrix entries. Explicitly, the matrix under consideration is:

$$\begin{pmatrix} 1 & a_{12} & a_{13} & a_{14} \\ 0 &     1 & a_{23} & a_{24} \\ 0 &     0 &      1 & a_{34} \\ 0 &     0 &      0 &     1  \\ \end{pmatrix}$$

The subgroups mentioned in the table below are:

Grouping by conjugacy class sizes
This follows by computing from the table in the previous section.

Conjugacy classes with respect to the general linear group
If we consider the action of the general linear group by conjugation, then there is considerable fusion of conjugacy classes. Specifically, there are only five equivalence classes, corresponding to the set of unordered integer partitions of 4 describing the possible Jordan block decompositions.

Below is a summary of the information:

Order statistics
The order statistics can be computed from the information in the section. The computations are below:

Case $$p \ge 5$$
In this case, all the non-identity elements have order $$p$$.