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) does not always satisfy a particular subgroup property (i.e., quotient-powering-faithful subgroup)
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Statement

It is possible to have a nilpotent group $G$ such that the abelianization of $G$ is a rationally powered group (which in this case means it is a vector space over $\mathbb{Q}$) but such that $G$ is not rationally powered over any prime. In fact, we can select $G$ to be such that it has $p$-torsion for every prime $p$.

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

Proof

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

Let $G$ be the quotient group of $UT(3,\mathbb{Q})$ (the unitriangular matrix group of degree three over the field of rational numbers) by a subgroup $\mathbb{Z}$ inside its center (which is a copy of $\mathbb{Q}$). Explicitly, we can think of $G$ as matrices of the form: $\{ \begin{pmatrix} 1 & a_{12} & \overline{a_{13}} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix} \mid a_{12},a_{23} \in \mathbb{Q}, \overline{a_{13}} \in \mathbb{Q}/\mathbb{Z} \}$

with the matrix multiplication defined as: $\begin{pmatrix} 1 & a_{12} & \overline{a_{13}} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix}\begin{pmatrix} 1 & b_{12} & \overline{b_{13}} \\ 0 & 1 & b_{23} \\ 0 & 0 & 1 \\\end{pmatrix} = \begin{pmatrix} 1 & a_{12} + b_{12} & \overline{a_{12}b_{23}} + \overline{a_{13}} + \overline{b_{13}} \\ 0 & 1 & a_{23} + b_{23} \\ 0 & 0 & 1 \\\end{pmatrix}$

where $\overline{a_{12}b_{23}}$ is understood to be the image of $a_{12}b_{23}$ under the quotient map $\mathbb{Q} \to \mathbb{Q}/\mathbb{Z}$.

The center of $G$ coincides with its derived subgroup, and is: $\left \{ \begin{pmatrix} 1 & 0 & \overline{a_{13}} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix} \mid \overline{a_{13}} \in \mathbb{Q}/\mathbb{Z} \right \}$

The inner automorphism group and the abelianization are therefore both isomorphic to $\mathbb{Q} \times \mathbb{Q}$, which is rationally powered. However, the group as a whole has $p$-torsion for all primes $p$.