# Special linear group:SL(2,Z)

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## 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 special linear group over integers and hence of a special linear group. It is also a special case of a special linear group of degree two.

The group also has the following equivalent descriptions:

### Structures

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

## Arithmetic functions

Function Value Explanation
order infinite (countable)
exponent infinite (countable)

## GAP implementation

The group can be defined using GAP's SpecialLinearGroup as:

SL(2,Integers)