presentation of a group
A presentation of a group is a description of 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 is said to generate if every element of can be specified as a product of the , and of their inverses. A relation is a word over the alphabet consisting of the generators and their inverses, with the property that it multiplies out to the identity in . A set of relations is said to be defining, if all relations in can be given as a product of the , their inverses, and the -conjugates of these.
The standard notation for the presentation of a group is
meaning that is generated by generators , subject to relations . Equivalently, one has a short exact sequence of groups
where denotes the free group generated by the , and where is the smallest normal subgroup containing all the . By the Nielsen-Schreier Theorem, the kernel is itself a free group, and hence we assume without loss of generality that there are no relations among the relations.
This means that a finite symmetric group is a Coxeter group.
|Title||presentation of a group|
|Date of creation||2013-03-22 12:23:23|
|Last modified on||2013-03-22 12:23:23|
|Last modified by||rmilson (146)|
|Defines||generators and relations|