Presentation theory of symmetric groups

This article discusses the presentations used for symmetric groups, specifically, for symmetric groups on finite sets.

Coxeter presentations: using transpositions as a generating set

Further information: Transpositions generate the finitary symmetric group, Transpositions of adjacent elements generate the symmetric group on a finite set, Symmetric group on a finite set is a Coxeter group

Suppose $G$ is the symmetric group on the set $\{1,2,3,\dots ,n+1\}$. Then, $G$ is isomorphic to a Coxeter group, with the following Coxeter presentation:

$G\cong \langle s_{i}\mid s_{i}^{2}=1\forall 1\leq i\leq n,(s_{i}s_{i+1})^{3}=1\forall 1\leq i\leq n-1,(s_{i}s_{j})^{2}=1\forall 1\leq i<j\leq n,|i-j|>1\rangle$.

This presentation of the symmetric group on $n+1$ letters has $n$ generators and $n(n+1)/2$ relations.