Polycyclic group

Remarks

Equivalently, $G$ is solvable and every subgroup of $G$ is finitely generated. The number of infinite cyclic factors does not depend on the series; it is the Hirsch length of $G$. Example: $\Z^2$ is polycyclic, with the series $0 \le \Z \times 0 \le \Z^2$ whose factors are both isomorphic to $\Z$; its Hirsch length is $2$. Non-example: the additive group $\Q$ is abelian, hence solvable, but it is not polycyclic. A polycyclic group is generated by lifts of generators of its factors, so it is finitely generated, while $\Q$ is not finitely generated. Every finite solvable group is polycyclic, and every supersolvable group is polycyclic.