---
tag: 0MBV2VHMXA
kind: concept
name: Generators of a group
aliases: [generating-set-of-a-group]
msc: [20F05]
uses: [0RG9SPQSWM]
sources:
  - url: https://en.wikipedia.org/wiki/Generating_set_of_a_group
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Generating_set_of_a_group
    locator: Examples
same_as:
  wikidata: Q734209
  wikipedia: Generating set of a group
---

## Summary

A subset $S$ of a [[group|G5S1M9]] $G$ generates $G$ when the [[subgroup generated by $S$|0RP7VWQ0YN]] is all of $G$.
Then each element of $G$ is a finite product of elements of $S$ and of their inverses. The elements of $S$ are called generators of $G$.

## Remarks

The set $\{1\}$ generates $(\mathbb{Z}, +)$, and so does the set $\{2, 3\}$.
Each group is generated by the set of all its elements.
A group generated by one element is a cyclic group, and a group generated by a finite set is a finitely generated group.

## Explanations

### L1 human

Generators are building blocks for a group: you can reach each element from them with products and inverses.
A small set of generators gives a short description of a big group.
For example, all symmetries of a square come from one rotation and one reflection.