presentation

A presentation of a group is a description of a set I and a subset R of the free group F(I) generated by I, written 〈(x_i)_(i element I)|(r)_(r element R)〉, where r = 1 (the identity element) is often written in place of r. A group presentation defines the quotient group of the free group F(I) by the normal subgroup generated by R, which is the group generated by the generators x_i subject to the relations r element R. Examples of group presentations include the following. 1. The presentation 〈x, y|x^2 = 1, y^n = 1, (x y)^2 = 1〉 defines a group, isomorphic to the dihedral group D_n of finite order 2n, which is the group of symmetries of a regular n-gon.