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
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As a reduced free group
As a matrix group
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:
The group has the structure of an arithmetic group.
|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 , and then goes to the whole group. Each of the quotient groups is isomorphic to .|
|polycyclic breadth||3||We can use a polycyclic series that starts with the center, then goes to the subgroup , and then goes to the whole group. Each of the quotient groups is isomorphic to .|
|group of nilpotency class two||Yes|