---
tag: TXYS4HF8JM
kind: concept
name: Free-by-cyclic group
aliases: [free-by-cyclic]
sources:
  - wikipedia: en:10377468@r1359409430
    locator: Free-by-cyclic group
    note: adopted
    mention: wikipedia://en/10377468
    sections: [Summary]
same_as:
  wikidata: Q5499608
  wikipedia: Free-by-cyclic group
inventory: [FG8GMNXS7J]
---

## Summary

In group theory, especially, in geometric group theory, the class of free-by-cyclic groups have been deeply studied as important examples. A group $G$ is said to be **free-by-cyclic** if it has a [[free|020Z5SEN27]] [[normal subgroup|0CB0DX5BVV]] $F$ such that the [[quotient group|0KYX2N69J0]] $G/F$ is [[cyclic|0B85Q41RP6]]. In other words, $G$ is free-by-cyclic if it can be expressed as a group extension of a free group by a cyclic group (NB there are two conventions for 'by'). Usually, we assume $F$ is finitely generated and the quotient is an infinite cyclic group. Equivalently, we can define a free-by-cyclic group constructively: if $\varphi$ is an automorphism of $F$, the semidirect product $F \rtimes_\varphi \mathbb{Z}$ is a free-by-cyclic group.