Projective special linear group:PSL(2,Z)

Definition
The group $$PSL(2,\Z)$$, also sometimes called the modular group, is defined in the following equivalent ways:


 * 1) It is the member of family::projective special linear group of degree two over the ring of integers. In other words, it is the quotient of the special linear group:SL(2,Z) by the subgroup $$\pm I$$.
 * 2) It is the defining ingredient::inner automorphism group of defining ingredient::braid group:B3, i.e., the quotient of $$B_3$$ by its center.
 * 3) It is the free product of defining ingredient::cyclic group:Z2 and defining ingredient::cyclic group:Z3.

Definition by presentation
The group can be defined by the following presentation, where $$1$$ denotes the identity element:


 * 1) As a projective special linear group: $$\langle S,T \mid S^2 = 1, (ST)^3 = 1 \rangle$$ where $$S$$ is the image of the matrix $$\begin{pmatrix} 0 & -1 \\ 1 & 0 \\\end{pmatrix}$$ and $$T$$ is the image of the matrix $$\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}$$.
 * 2) As a matrix group: $$\langle x,y \mid x^2 = y^3 = 1 \rangle$$