# Nilpotent with rationally powered center not implies rationally powered

From Groupprops

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., center) doesnotalways satisfy a particular subgroup property (i.e., powering-faithful subgroup)

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

## Statement

It is possible to have a nilpotent group such that the center is rationally powered, but such that is not powered over *any* prime.

In particular, it is possible to have a nilpotent group such that the center is not a powering-faithful subgroup of .

## Related facts

## Related facts

### Dual fact

The dual fact to this is nilpotent group with rationally powered abelianization need not be rationally powered.

### Similar facts

- Nilpotent and torsion-free not implies torsion-free abelianization
- Nilpotent group with rationally powered abelianization need not be rationally powered

## Proof

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 isomorphic to the group of rational numbers, hence is powered over every prime. However, is not powered over any prime (we can see this from the fact that is not -divisible for any prime ).