# Nilpotent and torsion-free not implies torsion-free abelianization

This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., derived subgroup) doesnotalways satisfy a particular subgroup property (i.e., quotient-torsion-freeness-closed subgroup)

View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions

## Statement

It is possible to have a torsion-free nilpotent group such that the abelianization of has -torsion for *every* prime .

In particular, this shows that the derived subgroup in a nilpotent group is not necessarily a quotient-torsion-freeness-closed subgroup.

## Related facts

### Dual fact

Somewhat surprisingly, the dual fact to this is not true. The dual fact, if true, would state that the center of a divisible nilpotent group need not be divisible (and in particular, that the center need not be divisibility-closed in a nilpotent group). This is false. In fact, upper central series members are completely divisibility-closed in nilpotent group.

### Converse

## Proof

`Further information: central product of UT(3,Z) and Q`

Let be the central product of unitriangular matrix group:UT(3,Z) with the group of rational numbers, where the center of the former is identified with a copy of in the latter. Then,

- is torsion-free.
- is isomorphic to , and is isomorphic to . This has -torsion for all primes .