# nLab permutation group

### Context

#### Group Theory

group theory

Classical groups

Finite groups

Group schemes

Topological groups

Lie groups

Super-Lie groups

Higher groups

Cohomology and Extensions

Related concepts

# Contents

## Introduction

A permutation group is, roughly speaking, a set of permutations which is closed under composition and which includes the identity permutation. Hence these are the subgroups of symmetric groups. Permutation groups are of historical significance: they were the first groups to be studied.

By Cayley's theorem, all (discrete) groups are in fact isomorphic to permutation groups.

## Formal definition and examples

To give a formal definition of a permutation group, we make use of the symmetric group.

###### Definition

Let $X$ be a finite set. A permutation group on $X$ is a subgroup of the symmetric group on $X$.

###### Example

The alternating group on a finite set $X$ is a permutation group on $X$.

###### Example

Let $X$ be a finite set with $n$ elements. Let $i$ be a fixed isomorphism $X \rightarrow \{1, \ldots, n \}$. The group of rotation permutations of $X$ with respect to $i$ is a permutation group on $X$.

Last revised on December 31, 2018 at 23:20:59. See the history of this page for a list of all contributions to it.