---
tag: 9RBA56NXX3
kind: concept
name: Polycyclic group
msc: [20F16, 20F05]
specializes: [0FXJ7YFMMX]
sources:
  - url: https://en.wikipedia.org/wiki/Polycyclic_group
same_as:
  wikidata: Q1658427
  wikipedia: Polycyclic group
---

## Summary

A polycyclic group is a group with a subnormal series whose factors are all cyclic.
Equivalently, it is a solvable group in which every subgroup is finitely generated.
Polycyclic groups form a well-behaved class of infinite groups: they are finitely presented, and many algorithmic questions about them can be decided.

## Explanations

### L1 human

A polycyclic group is built by stacking cyclic groups one on top of another.
Each layer is generated by a single element, so the whole group is described by finitely many elements and the way they act on each other.
The group $\Z^2$ is polycyclic: the series $0 \le \Z \times 0 \le \Z^2$ has factors $\Z$ and $\Z$.
Every finite solvable group is polycyclic, but the additive group of the rational numbers is not, since it is not finitely generated.