The <lamplighter group>
$$
C_2\wr\mathbb Z=\left(\bigoplus_{n\in\mathbb Z}C_2\right)\rtimes\mathbb Z
$$
is generated by one lamp switch and one translation. It is <metabelian group>[metabelian], hence solvable. Its base subgroup $\bigoplus_{\mathbb Z}C_2$ is not finitely generated. Every subgroup of a polycyclic group is finitely generated, so the lamplighter group is not polycyclic.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 104 / 2 / a / 3 / Solution

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A group is metabelian when its <commutator subgroup> is <abelian group>[abelian], equivalently when its derived length is at most two.

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