# Quotient of UT(3,Q) by a central Z

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## Definition

The group described here is a quotient group of unitriangular matrix group:UT(3,Q) by a central subgroup isomorphic to the group of integers, which we can think of as a Z in Q inside the center, which is a copy of . Explicitly, it is matrices of the form:

with the matrix multiplication defined as:

where is understood to be the image of under the quotient map .

## Arithmetic functions

Function | Value | Similar groups | Explanation |
---|---|---|---|

order | countably infinite | ||

exponent | infinite (elements of infinite order) | ||

nilpotency class | 2 | ||

derived length | 2 |

## Group properties

Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|

abelian group | any two elements commute | No | ||

group of nilpotency class two | Yes | |||

metabelian group | Yes | |||

torsion-free group | No |