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

The Integer Heisenberg groupis generated by the matrices with $(a,b,c)=(1,0,0)$ and $(0,1,0)$. Their group commutator is the nonidentity central matrix with $(a,b,c)=(0,0,1)$. Thus it is nonabelian and nilpotent of class two, while being finitely generated.

The countable direct sumis abelian and therefore nilpotent of class one. It is not finitely generated, whereas every polycyclic group is finitely generated. Hence it is nilpotent but not polycyclic.

The lamplighter groupis generated by one lamp switch and one translation. It is metabelian, hence solvable. Its base subgroup ${⨁}_{\mathbb{Z}}{C}_2$ is not finitely generated. Every subgroup of a polycyclic group is finitely generated, so the lamplighter group is not polycyclic.