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

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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_pq 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 \ge 5: order 1 (1 element), order p (q^6 - 1 elements)
exponent p^2 if p < 5
p if p \ge 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} \ne 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
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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 \ge 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 \ge 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