A presentation of a group is a description of a set and a subset of the free group generated by , 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 order , which is the group of symmetries of a regular -gon.
2. The fundamental group of a surface of genus has the presentation
3. Coxeter groups.
4. Braid groups.