# presentation of a group

A presentation of a group $G$ is a description of $G$ in terms of generators and relations (sometimes also known as relators). We say that the group is finitely presented, if it can be described in terms of a finite number of generators and a finite number of defining relations. A collection of group elements $g_{i}\in G,\;i\in I$ is said to generate $G$ if every element of $G$ can be specified as a product of the $g_{i}$, and of their inverses. A relation is a word over the alphabet consisting of the generators $g_{i}$ and their inverses, with the property that it multiplies out to the identity in $G$. A set of relations $r_{j},\;j\in J$ is said to be defining, if all relations in $G$ can be given as a product of the $r_{j}$, their inverses, and the $G$-conjugates of these.

The standard notation for the presentation of a group is

 $G=\langle g_{i}\mid r_{j}\rangle,$

meaning that $G$ is generated by generators $g_{i}$, subject to relations $r_{j}$. Equivalently, one has a short exact sequence of groups

 $1\to N\to F[I]\to G\to 1,$

where $F[I]$ denotes the free group generated by the $g_{i}$, and where $N$ is the smallest normal subgroup containing all the $r_{j}$. By the Nielsen-Schreier Theorem, the kernel $N$ is itself a free group, and hence we assume without loss of generality that there are no relations among the relations.

Example. The symmetric group on $n$ elements $1,\ldots,n$ admits the following finite presentation (Note: this presentation is not canonical. Other presentations are known.) As generators take

 $g_{i}=(i,i+1),\quad i=1,\ldots,n-1,$

the transpositions of adjacent elements. As defining relations take

 $(g_{i}g_{j})^{n_{i,j}}=\mathrm{id},\quad i,j=1,\ldots n,$

where

 $\displaystyle n_{i,i}$ $\displaystyle=1$ $\displaystyle n_{i,i+1}$ $\displaystyle=3$ $\displaystyle n_{i,j}$ $\displaystyle=2,\quad|j-i|>1.$

This means that a finite symmetric group is a Coxeter group.

 Title presentation of a group Canonical name PresentationOfAGroup Date of creation 2013-03-22 12:23:23 Last modified on 2013-03-22 12:23:23 Owner rmilson (146) Last modified by rmilson (146) Numerical id 20 Author rmilson (146) Entry type Definition Classification msc 20A05 Classification msc 20F05 Synonym presentation Synonym finite presentation Synonym finitely presented Related topic GeneratingSetOfAGroup Related topic CayleyGraph Defines generator Defines relation Defines generators and relations Defines relator