---
tag: 0HY15D6WTX
kind: definition
name: Finitely generated group
msc: [20F05]
defines: 082DWWXMRJ
uses: [0RG9SPQSWM]
sources:
  - url: https://en.wikipedia.org/wiki/Finitely_generated_group
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Finitely_generated_group
    locator: Examples
---

## Definition

A [[group|G5S1M9]] $G$ is finitely generated when there is a finite subset $S$ of $G$ with $\langle S \rangle = G$, where $\langle S \rangle$ is the [[subgroup generated by $S$|0RP7VWQ0YN]].

## Remarks

Each finite group is finitely generated, because it is generated by the finite set of all its elements.
The group $(\mathbb{Z}^n, +)$ is finitely generated by the $n$ standard basis vectors.
The group $(\mathbb{Q}, +)$ is not finitely generated: the subgroup generated by finitely many fractions has a common denominator, and it does not contain all fractions.

## Explanations

### L1 human

A group is finitely generated when a finite list of elements is enough to build all elements.
The group may still be infinite, as the integers show: one generator $1$ gives all integers.
The rational numbers need infinitely many generators.