---
tag: JNB1VM75XP
kind: definition
name: Polycyclic group
msc: [20F16, 20F05]
defines: 9RBA56NXX3
uses: [0ZZR8D7Y1Z, 0KVGH0J5B3, 082DWWXMRJ]
sources:
  - url: https://en.wikipedia.org/wiki/Polycyclic_group
---

## Definition

Let $G$ be a [[group|G5S1M9]].
The group $G$ is *polycyclic* when it has a [[subnormal series|0ZZR8D7Y1Z]]
$$\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$$
in which each factor $G_i/G_{i-1}$ is [[cyclic|0KVGH0J5B3]].

## Remarks

Equivalently, $G$ is [[solvable|0FXJ7YFMMX]] and every subgroup of $G$ is [[finitely generated|082DWWXMRJ]].
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|AWWM9RWETX]] is polycyclic.

## Explanations

### L1 human

A group is polycyclic when it can be built from finitely many cyclic layers, each normal in the next.