Special linear group:SL(2,Z)

Definition
The group $$SL(2,\mathbb{Z})$$ is defined as the group, under matrix multiplication, of $$2 \times 2$$ matrices over $$\mathbb{Z}$$, the ring of integers, having determinant $$1$$.

In other words, it is the group with underlying set:

$$\left \{ \begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mid a,b,c,d \in \mathbb{Z}, ad - bc = 1 \right \}$$

This is the degree two case of a member of family::special linear group over integers and hence of a member of family::special linear group. It is also a special case of a member of family::special linear group of degree two.

The group also has the following equivalent descriptions:


 * The defining ingredient::amalgamated free product of defining ingredient::cyclic group:Z4 and defining ingredient::cyclic group:Z6 over amalgamated subgroup cyclic group:Z2 (living as Z2 in Z4 and Z2 in Z6 respectively).

Definition by presentation
The group can be defined by any of the following presentations (here, $$1$$ denotes the identity element):


 * From the amalgamated free product definition: $$\langle x,y \mid x^4 = 1, x^2 = y^3 \rangle$$
 * From the amalgamated free product definition: $$\langle x,y \mid x^4 = 1, x^2 = y^3 \rangle$$

Structures
Thinking of $$SL(2,\mathbb{Z})$$ as a group of matrices, we see that it is an example of an has structure of::arithmetic group.

Facts

 * Sanov subgroup in SL(2,Z) is free of rank two
 * The homomorphism $$SL(2,\mathbb{Z}) \to SL(2,\mathbb{Z}/n\mathbb{Z})$$ is surjective for any natural number $$n$$.