presentation of a group

A presentationMathworldPlanetmathPlanetmathPlanetmath 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 generatorsPlanetmathPlanetmathPlanetmath and a finite number of defining relations. A collectionMathworldPlanetmath of group elements giG,iI is said to generate G if every element of G can be specified as a productMathworldPlanetmathPlanetmathPlanetmath of the gi, and of their inversesMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. A relationMathworldPlanetmathPlanetmath is a word over the alphabet consisting of the generators gi and their inverses, with the property that it multiplies out to the identityPlanetmathPlanetmath in G. A set of relations rj,jJ is said to be defining, if all relations in G can be given as a product of the rj, their inverses, and the G-conjugates of these.

The standard notation for the presentation of a group is


meaning that G is generated by generators gi, subject to relations rj. Equivalently, one has a short exact sequenceMathworldPlanetmath of groups


where F[I] denotes the free groupMathworldPlanetmath generated by the gi, and where N is the smallest normal subgroup containing all the rj. 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 groupMathworldPlanetmathPlanetmath on n elements 1,,n admits the following finite presentation (Note: this presentation is not canonical. Other presentations are known.) As generators take


the transpositionsMathworldPlanetmath of adjacent elements. As defining relations take



ni,i =1
ni,i+1 =3
ni,j =2,|j-i|>1.

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

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