presentation of a group

A presentation of a group $G$ is a description of $G$ in terms ofgenerators and relations (sometimes also known as relators).We say that the group is finitelypresented, if it can be described in terms of a finite number ofgenerators and a finite number of defining relations. A collection ofgroup elements $g_{i}\\in G,\\;i\\in I$ is said to generate $G$ if everyelement of $G$ can be specified as a product of the $g_{i}$ , and of theirinverses. A relation is a word over the alphabet consisting of thegenerators $g_{i}$ and their inverses, with the property that itmultiplies 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 givenas a product of the $r_{j}$ , their inverses, and the $G$ -conjugates ofthese.

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 torelations $r_{j}$ . Equivalently, one has a short exact sequence ofgroups

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

where $F[I]$ denotes the free groupgenerated by the $g_{i}$ , and where $N$ is the smallest normal subgroupcontaining all the $r_{j}$ . By the Nielsen-Schreier Theorem, the kernel $N$ is itself a free group, and hence we assume without loss of generalitythat 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 isnot 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