---
tag: 0AQNX7G443
kind: statement
role: proposition
name: The symmetric group is generated by transpositions
msc: [20B30]
uses: [0XK6FECMW5, 0RG9SPQSWM]
sources:
  - url: https://en.wikipedia.org/wiki/Symmetric_group
    locator: Generators and relations
  - url: https://en.wikipedia.org/wiki/Cyclic_permutation
    locator: Properties
---

## Statement

Let $n \geq 2$.
The [[symmetric group|0515A6D2Y1]] $S_n$ is [[generated|0RP7VWQ0YN]] by its [[transpositions|0FWVRHBRWM]].

## Remarks

This is a reformulation of the [[decomposition into transpositions|0XK6FECMW5]].
The adjacent transpositions $(1\ 2), (2\ 3), \dots, (n-1\ n)$ already generate $S_n$.
The two elements $(1\ 2)$ and $(1\ 2\ \dots\ n)$ also generate $S_n$.

## Explanations

### L1 human

Each permutation is a product of swaps, so the swaps generate the whole symmetric group.
You do not even need all swaps.
Swaps of neighbours are enough, as in a bubble sort.