Unitriangular matrix group:UT(3,Z)

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Definition

As a reduced free group

Abstractly, this group is a free class two group on a generating set of size two. Hence, it is a reduced free group.

As a matrix group

This group, denoted $UT(3,\mathbb{Z})$ or $U(3,\mathbb{Z})$, is defined as the unitriangular matrix group of degree three over the ring of integers. Explicitly, it is the group, under multiplication:

$\left \{ \begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix} \mid a_{12},a_{13},a_{23} \in \mathbb{Z} \right \}$

The group is also sometimes called the integral Heisenberg group.

Definition by presentation

The group can be defined by means of the following presentation:

$\langle x,y,z \mid [x,y] = z, xz = zx, yz = zy \rangle$

We can relate this with the matrix group definition by setting:

$x = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix}, \qquad y = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\\end{pmatrix}, \qquad z = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix}$

Structures

The group has the structure of an arithmetic group.

Arithmetic functions

Function	Value	Similar groups	Explanation
nilpotency class	2		The derived subgroup and center are both equal to the subgroup $\left \{ \begin{pmatrix} 1 & 0 & a_{13} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix} \mid a_{13} \in \mathbb{Z} \right \}$
derived length	2		Follows from nilpotency class being 2.
Frattini length	2		The Frattini subgroup also coincides with the derived subgroup and center, and it is isomorphic to the group of integers, which is a Frattini-free group.
Hirsch length	3		We can use a polycyclic series that starts with the center, then goes to the subgroup $\left \{ \begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix} \mid a_{12}, a_{13} \in \mathbb{Z} \right \}$, and then goes to the whole group. Each of the quotient groups is isomorphic to $\mathbb{Z}$.
polycyclic breadth	3		We can use a polycyclic series that starts with the center, then goes to the subgroup $\left \{ \begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix} \mid a_{12}, a_{13} \in \mathbb{Z} \right \}$, and then goes to the whole group. Each of the quotient groups is isomorphic to $\mathbb{Z}$.

Group properties

Property	Satisfied?	Explanation
abelian group	No
nilpotent group	Yes
group of nilpotency class two	Yes
metacyclic group	No
polycyclic group	Yes
metabelian group	Yes
supersolvable group	Yes