Central product of UT(3,Z) and Q

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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

This group is defined as an external central product of unitriangular matrix group:UT(3,Z) with the group of rational numbers $\mathbb {Q}$, where we identify the full center of the first group with the subgroup $\mathbb {Z} \leq \mathbb {Q}$ in the latter (see Z in Q).

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		Yes