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