Nilpotent group with rationally powered abelianization need not be rationally powered

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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)
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 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.

Related facts

Opposite facts

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.