Finitely generated 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 generated solvable group) need not satisfy the second group property (i.e., polycyclic group)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about finitely generated solvable group|Get more facts about polycyclic group

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 generated solvable group) need not satisfy the second group property (i.e., Noetherian group)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about finitely generated solvable group|Get more facts about Noetherian group

Statement

It is possible to have a finitely generated solvable group that is not a polycyclic group, and hence, not a Noetherian group, i.e., it has a subgroup that is not finitely generated.

Related facts

• Finitely generated not implies Noetherian

Proof

A general construction using a restricted wreath product

Let $A$ be a nontrivial finitely generated solvable group. Let $G$ be the restricted external wreath product of $A$ and the group of integers $\mathbb {Z}$ acting regularly. In other words, $G$ is the external semidirect product of $H$ and $\mathbb {Z}$, where $H$ is the restricted external direct product of countably many copies of $A$ and $\mathbb {Z}$ acts on the coordinates by a shift of one.

Some examples based on the general construction and otherwise

• The wreath product of group of integers with group of integers is a finitely generated solvable group that is not polycyclic. The base of the semidirect product here is a countable restricted direct power of the group of integers, which is not finitely generated.
• Baumslag-Solitar group:BS(1,2) is a finitely generated solvable group that is not polycyclic. The derived subgroup of this group is isomorphic to the additive group of 2-adic rational numbers, which is not finitely generated.