---
tag: 0515A6D2Y1
kind: concept
name: Symmetric group
msc: [20B30]
sources:
  - url: https://en.wikipedia.org/wiki/Symmetric_group
    locator: lead section
same_as:
  wikidata: Q849512
  wikipedia: Symmetric group
---

## Summary

The symmetric group on a set $X$ is the group of all bijections from $X$ to $X$, with composition as the operation.
For $X = \{1, \dots, n\}$ it is written $S_n$, and its elements are the permutations of $n$ symbols.