# Contents

## Idea

A permutation is an automorphism in Set. More explicitly, a permutation of a set $X$ is an invertible function from $X$ to itself.

## Definition

The group of permutations of $X$ (that is the automorphism group of $X$ in $Set$) is the symmetric group (or permutation group) on $X$. This group may be denoted $S_X$, $\Sigma_X$, or $X!$. When $X$ is the finite set $[n]$ with $n$ elements, one typically writes $S_n$ or $\Sigma_n$; note that this group has $n!$ elements.

In combinatorics, one often wants a slight generalisation. Given a natural number $r$, an $r$-permutation from $X$ is an injective function from $[r]$ to $X$, that is a list of $r$ distinct elements of $X$. Then an $n$-permutation from $[n]$ is the same as a permutation of $[n]$. (That an injective function from $X$ to itself must be invertible characterises $X$ as a Dedekind-finite set.)

## Properties

### Relation to the field with one element

One may regard the symmetric group $S_n$ as the general linear group in dimension $n$ on the field with one element $GL(n,\mathbb{F}_1)$.

Revised on August 9, 2013 13:39:32 by Urs Schreiber (82.113.98.165)