A group presentation is a description of a set 
of generators and a subset 
of the free group 
generated by 
. It is written 
, where 
(the identity element)
is often written in place of 
. A group presentation defines the quotient
group of the free group 
by the normal subgroup
generated by 
,
which is the group generated by the generators 
subject to the relations 
.
Examples of group presentations include the following.
1. The presentation 
defines a group, isomorphic
to the dihedral group 
of finite group
order 
,
which is the group of symmetries
of a regular polygon with 
sides.
2. The fundamental group of a closed connected orientable surface of genus 
has the presentation
![<x_1,y_1,x_2,...,x_g,y_g|[x_1,y_1][x_2,y_2]...[x_g,y_g]=1>.](/images/equations/GroupPresentation/NumberedEquation1.svg) |
3. Coxeter groups.
4. Braid groups.
