Unitriangular matrix group:UT(3,Z)
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Contents
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 or
, is defined as the unitriangular matrix group of degree three over the ring of integers. Explicitly, it is the group, under multiplication:
The group is also sometimes called the integral Heisenberg group.
Definition by presentation
The group can be defined by means of the following presentation:
We can relate this with the matrix group definition by setting:
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 ![]() | |
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 ![]() ![]() | |
polycyclic breadth | 3 | We can use a polycyclic series that starts with the center, then goes to the subgroup ![]() ![]() |
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 |