---
tag: ZDJQRKPWGA
kind: concept
name: Permutation group
msc: [20B05, 20B07]
specializes: [0X1JXT47GA]
sources:
  - url: https://en.wikipedia.org/wiki/Permutation_group
same_as:
  wikidata: Q1412905
  wikipedia: Permutation group
---

## Summary

A permutation group is a [[subgroup|0X1JXT47GA]] of the [[symmetric group|0515A6D2Y1]] of a set, that is, a set of [[permutations|05RWFKVTJ7]] closed under composition and inverses.
Groups first arose as permutation groups, in the study of the roots of polynomial equations.
By [[Cayley's theorem|0151VYVX27]], every group is isomorphic to a permutation group.
Every [[group action|0WCY42EV9V]] gives a permutation group, the image of the group in the symmetric group of the set.

## Explanations

### L1 human

A permutation group is a collection of ways to shuffle a set of objects that is closed under doing one shuffle after another and under undoing a shuffle.
The symmetries of a square, viewed as shuffles of its four corners, are a good picture.
If the corners are labelled $1, 2, 3, 4$ in order around the square, the quarter turn is the shuffle $1 \to 2 \to 3 \to 4 \to 1$.
The eight symmetries of the square give eight of the $24$ possible shuffles of the corners, and they form a permutation group.