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

This article gives specific information, namely, element structure, about a family of groups, namely: unitriangular matrix group of degree four.
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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 $\log _{p}q$ by $r$. Further, we denote the group as $UT(4,q)$.

Summary

Item	Value
number of conjugacy classes	$2q^{3}+q^{2}-2q$ equals number of irreducible representations. See number of irreducible representations equals number of conjugacy classes, linear representation theory of unitriangular matrix group of degree four over a finite field
order	$q^{6}$ Follows from the general formula, order of $UT(n,q)$ is $q^{n(n-1)/2}=q^{6}$
conjugacy class size statistics	1 ($q$ times), $q$ ($q^{2}-1$ times), $q^{2}$ ($q^{3}+q^{2}-2q$ times), $q^{3}$ ($q^{3}-q^{2}-q+1$ times)
order statistics	Case $p=2$: order 1 (1 element), order 2 ($2q^{4}-q^{2}-1$ elements), order 4 ($(q^{3}-q)^{2}$ elements) Case $p=3$: order 1 (1 element), order 3 ($q^{6}-1-q^{3}(q-1)^{3}$ elements), order 9 ($q^{3}(q-1)^{3}$ elements) Case $p\geq 5$: order 1 (1 element), order $p$ ($q^{6}-1$ elements)
exponent	$p^{2}$ if $p<5$ $p$ if $p\geq 5$ The exponent depends only on $p$, not on $q$.

Conjugacy class structure

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:

Subgroup	Visual description	Condition	Order
center	${\begin{pmatrix}1&0&0&a_{14}\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{pmatrix}}$	$a_{12}=a_{13}=a_{23}=a_{24}=a_{34}=0$	$q$
derived subgroup	${\begin{pmatrix}1&0&a_{13}&a_{14}\\0&1&0&a_{24}\\0&0&1&0\\0&0&0&1\\\end{pmatrix}}$	$a_{12}=a_{23}=a_{34}=0$	$q^{3}$
unique abelian subgroup of maximum order	${\begin{pmatrix}1&0&a_{13}&a_{14}\\0&1&a_{23}&a_{24}\\0&0&1&0\\0&0&0&1\\\end{pmatrix}}$	$a_{12}=a_{34}=0$	$q^{4}$

Nature of conjugacy class	Jordan block size decomposition	Minimal polynomial	Size of conjugacy class	Number of such conjugacy classes	Total number of elements	Order of elements in each such conjugacy class	Type of matrix (constraints on $a_{ij},i<j$)
identity element	1 + 1 + 1 + 1	$t-1$	1	1	1	1	all the $a_{ij},i<j$ are zero
non-identity element, but central (has Jordan blocks of size 1,1,2 respectively)	2 + 1 + 1	$(t-1)^{2}$	1	$q-1$	$q-1$	$p$	$a_{14}\neq 0$, all the others are zero
non-central but in derived subgroup, has Jordan blocks of size 1,1,2	2 + 1 + 1	$(t-1)^{2}$	$q$	$2(q-1)$	$2q(q-1)$	$p$	$a_{12}=a_{23}=a_{34}=0$ Among $a_{13}$ and $a_{24}$, exactly one of them is nonzero. $a_{14}$ may be zero or nonzero
non-central but in derived subgroup, Jordan blocks of size 2,2	2 + 2	$(t-1)^{2}$	$q$	$(q-1)^{2}$	$q(q-1)^{2}$	$p$	$a_{12}=a_{23}=a_{34}=0$ Both $a_{13}$ and $a_{24}$ are nonzero. $a_{14}$ may be zero or nonzero
outside derived subgroup, inside unique abelian subgroup of maximum order, with Jordan blocks of size 1,1,2	2 + 1 + 1	$(t-1)^{2}$	$q^{2}$	$q-1$	$q^{2}(q-1)$	$p$	$a_{12}=a_{34}=a_{14}=0$ $a_{23}$ is nonzero $a_{13}$ and $a_{24}$ are arbitrary
outside derived subgroup, inside unique abelian subgroup of maximum order, with Jordan blocks of size 2,2	2 + 2	$(t-1)^{2}$	$q^{2}$	$(q-1)^{2}$	$q^{2}(q-1)^{2}$	$p$	$a_{12}=a_{34}=0$ $a_{23}$ and $a_{14}$ are both nonzero $a_{13}$ and $a_{24}$ are arbitrary
outside abelian subgroup of maximum order, Jordan blocks of size 1,1,2	2 + 1 + 1	$(t-1)^{2}$	$q^{2}$	$2(q-1)$	$2q^{2}(q-1)$	$p$	Two subcases: Case 1: $a_{12}=a_{23}=a_{13}=0$, $a_{34}$ nonzero, $a_{14},a_{24}$ arbitrary Case 2: $a_{23}=a_{24}=a_{34}=0$, $a_{12}$ nonzero, $a_{13},a_{14}$ arbitrary
outside abelian subgroup of maximum order, Jordan blocks of size 2,2	2 + 2	$(t-1)^{2}$	$q^{2}$	$(q-1)^{2}$	$q^{2}(q-1)^{2}$	$p$	$a_{12},a_{34}$ both nonzero $a_{23}=0$ $a_{14},a_{24}$ arbitrary $a_{13}$ uniquely determined by other values
outside abelian subgroup of maximum order, Jordan blocks of size 1,3, with centralizer of order $q^{4}$	3 + 1	$(t-1)^{3}$	$q^{2}$	$(q-1)^{2}(q+1)$	$q^{2}(q-1)^{2}(q+1)$	$p$ if $p$ odd 4 if $p=2$	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
outside abelian subgroup of maximum order, Jordan blocks of size 1,3, with centralizer of order $q^{3}$	3 + 1	$(t-1)^{3}$	$q^{3}$	$2(q-1)^{2}$	$2q^{3}(q-1)^{2}$	$p$ if $p$ odd 4 if $p=2$	Two subcases: Case 1: $a_{12},a_{23}$ nonzero, $a_{34}=0$, other entries arbitrary Case 2: $a_{23},a_{34}$ nonzero, $a_{12}=0$, other entries arbitrary
Jordan block of size 4	4	$(t-1)^{4}$	$q^{3}$	$(q-1)^{3}$	$q^{3}(q-1)^{3}$	$p^{2}$ if $p<5$ $p$ if $p\geq 5$	$a_{12},a_{23},a_{34}$ nonzero $a_{13},a_{14},a_{24}$ arbitrary
Total (--)	--	--	--	$2q^{3}+q^{2}-2q$	$q^{6}$	--	--

Grouping by conjugacy class sizes

This follows by computing from the table in the previous section.

Conjugacy class size	Total number of conjugacy classes of this size	Total number of elements	Cumulative number of conjugacy classes	Cumulative number of elements
1	$q$	$q$	$q$	$q$
$q$	$q^{2}-1$	$q^{3}-q$	$q^{2}+q-1$	$q^{3}$
$q^{2}$	$q^{3}+q^{2}-2q$	$q^{5}+q^{4}-2q^{3}$	$q^{3}+2q^{2}-q-1$	$q^{5}+q^{4}-q^{3}$
$q^{3}$	$q^{3}-q^{2}-q+1$	$q^{6}-q^{5}-q^{4}+q^{3}$	$2q^{3}+q^{2}-2q$ (total)	$q^{6}$ (total)

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:

Jordan block size decomposition (partition of 4)	Number of conjugacy classes of size 1 Number of elements in these	Number of conjugacy classes of size $q$ Number of elements in these	Number of conjugacy classes of size $q^{2}$ Number of elements in these	Number of conjugacy classes of size $q^{3}$ Number of elements in these	Total (number of conjugacy classes, elements)
1 + 1 + 1 + 1	1 1	0 0	0 0	0 0	1 1
2 + 1 + 1	$q-1$ $q-1$	$2(q-1)$ $2q(q-1)$	$3(q-1)$ $3q^{2}(q-1)$	0 0	$6(q-1)$ $(q-1)(3q^{2}+2q+1)$
2 + 2	0 0	$(q-1)^{2}$ $q(q-1)^{2}$	$2(q-1)^{2}$ $2q^{2}(q-1)^{2}$	0 0	$3(q-1)^{2}$ $q(q-1)^{2}(2q+1)$
3 + 1	0 0	0 0	$(q-1)^{2}(q+1)$ $q^{2}(q-1)^{2}(q+1)$	$2(q-1)^{2}$ $2q^{3}(q-1)^{2}$	$(q-1)^{2}(q+3)$ $q^{2}(q-1)^{2}(3q+1)$
4	0 0	0 0	0 0	$(q-1)^{3}$ $q^{3}(q-1)^{3}$	$(q-1)^{3}$ $q^{3}(q-1)^{3}$
Total (--)	$q$ $q$	$q^{2}-1$ $q^{3}-q$	$q^{3}+q^{2}-2q$ $q^{5}+q^{4}-2q^{3}$	$q^{3}-q^{2}-q+1$ $q^{6}-q^{5}-q^{4}+q^{3}$	$2q^{3}+q^{2}-2q$ $q^{6}$

Order statistics

The order statistics can be computed from the information in the #Conjugacy class structure section. The computations are below:

Case $p=2$

Order	List of conjugacy class sizes of elements with that order	Total number of conjugacy classes	Total number of elements
1	size 1 (1 time)	1	1
$p$ (i.e., 2)	size 1 ($q-1$ times), size $q$ ($q^{2}-1$ times), size $q^{2}$ ($(q-1)(2q+1)$ times)	$3(q^{2}-1)$	$2q^{4}-q^{2}-1$
$p^{2}$ (i.e., 4)	size $q^{2}$ ($(q-1)^{2}(q+1)$ times), size $q^{3}$ ($(q-1)^{2}(q+1)$ times)	$2(q-1)^{2}(q+1)$	$q^{2}(q-1)^{2}(q+1)^{2}=(q^{3}-q)^{2}=q^{6}-2q^{4}+q^{2}$
Total (--)	--	$2q^{3}+q^{2}-2q$	$q^{6}$

Case $p=3$

Order	List of conjugacy class sizes of elements with that order	Total number of conjugacy classes	Total number of elements
1	size 1 (1 time)	1	1
$p$ (i.e., 3)	size 1 ($q-1$ times), size $q$ ($q^{2}-1$ times), size $q^{2}$ ($q^{3}+q^{2}-2q$ times), size $q^{3}$ ($2(q-1)^{2}$ times)	$q^{3}+4q^{2}-5q$	$3q^{5}-3q^{4}+q^{3}-1$
$p^{2}$ (i.e., 9)	size $q^{3}$ ($(q-1)^{3}$ times)	$(q-1)^{3}$	$q^{3}(q-1)^{3}=(q^{2}-q)^{3}=q^{6}-3q^{5}+3q^{4}-q^{3}$
Total (--)	--	$2q^{3}+q^{2}-2q$	$q^{6}$

Case $p\geq 5$

In this case, all the non-identity elements have order $p$.

Order	List of conjugacy class sizes of elements with that order	Total number of conjugacy classes	Total number of elements
1	size 1 (1 time)	1	1
$p$	size 1 ($q-1$ times), size $q$ ($q^{2}-1$ times), size $q^{2}$ ($q^{3}+q^{2}-2q$ times), size $q^{3}$ ($q^{3}-q^{2}-q+1$ times)	$2q^{3}+q^{2}-2q-1$	$q^{6}-1$
Total (--)	--	$2q^{3}+q^{2}-2q$	$q^{6}$