nLab permutation group

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 $X$.