# Nilpotent group with rationally powered abelianization need not be rationally powered

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-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 abelianization of is a rationally powered group (which in this case means it is a vector space over ) but such that is not rationally powered over any prime. In fact, we can select to be such that it has -torsion for *every* prime .

This also shows that the derived subgroup of a nilpotent group need not be a quotient-powering-faithful subgroup.

## Related facts

### Opposite facts

- The group
*is*forced to be divisible for all primes that the abelianization is divisible by. See Nilpotent group is divisible by a prime iff its abelianization is and iff all lower central series quotients are.

## Proof

`Further information: quotient of UT(3,Q) by a central Z`

Let be the quotient group of (the unitriangular matrix group of degree three over the field of rational numbers) by a subgroup inside its center (which is a copy of ). Explicitly, we can think of as matrices of the form:

with the matrix multiplication defined as:

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

The center of coincides with its derived subgroup, and is:

The inner automorphism group and the abelianization are therefore both isomorphic to , which is rationally powered. However, the group as a whole has -torsion for all primes .