Central product of UT(3,Z) and Z identifying center with 2Z

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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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Definition

This group can be defined in the following equivalent ways:

  1. It is the central product of unitriangular matrix group:UT(3,Z) and the group of integers \mathbb{Z} where the subgroup 2\mathbb{Z} of \mathbb{Z} is identified with the center of UT(3,\mathbb{Z}).
  2. It is the following group of matrices under multiplication:

\left\{ \begin{pmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\\end{pmatrix} \mid a,c \in \mathbb{Z}, 2b \in \mathbb{Z} \right \}

This group is almost like unitriangular matrix group:UT(3,Z). In fact, UT(3,\mathbb{Z}) occurs as a subgroup of index two inside it. However, unlike UT(3,\mathbb{Z}), it is a CS-Baer Lie group, and hence can participate in the CS-Baer correspondence.

Arithmetic functions

Function Value Similar groups Explanation
nilpotency class 2
derived length 2
Frattini length 2
Hirsch length 3
polycyclic breadth 3

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