Presentation theory of symmetric group:S3

This article discusses the typical presentations used for the symmetric group of degree three.

The dihedral presentation: rotation and reflection
One presentation of the symmetric group of degree three is by viewing it as a dihedral group of order six. Here, the two generating elements are a rotation (which could be any $$3$$-cycle) and a reflection (which could be any transposition). The presentation is then:

$$\langle a,x \mid a^3 = x^2 = 1, xax^{-1} = a^2 \rangle$$.

The dihedral group: two reflections
Another presentation of the symmetric group of degree three is by viewing it as a dihedral group, but now considering it to be generated by two reflections:

$$\langle x,y \mid x^2 = y^2 = 1, (xy)^3 = e\rangle$$.

A triangle group presentation
The two presentation given above are related by $$y = ax$$. Another presentation, which generalizes to the presentation used for von Dyck groups, puts three generators with four relations:

$$\langle a,b,c \mid a^2 = b^2 = c^3 = abc = 1 \rangle$$.