Finitely presented and solvable not implies polycyclic

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finitely presented solvable group) need not satisfy the second group property (i.e., polycyclic group)
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Statement

It is possible to have a finitely presented solvable group $G$ -- a group $G$ that is both a finitely presented group and a solvable group -- such that $G$ is not a polycyclic group. In particular, this means that $G$ is not a Noetherian group, i.e., it has a subgroup that is not finitely generated.

Related facts

• Finitely generated and solvable not implies finitely presented

Proof

Further information: Baumslag-Solitar group:BS(1,2)

The group $G=BS(1,2)$, the Baumslag-Solitar group with parameters $(1,2)$, works:

$\langle a,b\mid bab^{-1}=a^{2}\rangle$

The normal closure of $\langle a\rangle$ in this group, which is also the derived subgroup of $G$, is isomorphic to the group of 2-adic rationals, i.e., the group of all rationals with denominators powers of 2.