Unitriangular matrix group:UT(3,Z)

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

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