symmetric group

Let $X$ be a set. Let $\mathrm{Sym}(X)$ be the set of permutations of $X$ (i.e. the set of bijective functions from $X$ to itself). Then the act of taking the composition of two permutations induces a group structure on $\mathrm{Sym}(X)$. We call this group the symmetric group.

The group $\mathrm{Sym}(\{1,2,\mathrm{\dots },n\})$ is often denoted $S_n$ or ${𝔖}_n$.

$S_n$ is generated by the transpositions $\{(1,2),(2,3),\mathrm{\dots },(n-1,n)\}$, and by any pair of a 2-cycle and $n$-cycle.

$S_n$ is the Weyl group of the $A_{n-1}$ root system (and hence of the special linear group $S{L}_{n-1}$).

Title	symmetric group
Canonical name	SymmetricGroup
Date of creation	2013-03-22 12:01:53
Last modified on	2013-03-22 12:01:53
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	11
Author	bwebste (988)
Entry type	Definition
Classification	msc 20B30
Related topic	Group
Related topic	Cycle2
Related topic	CayleyGraphOfS_3
Related topic	Symmetry2