# 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

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

where

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