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:

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

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:

• The inner automorphism group of braid group:B3, i.e., the quotient of $B_3$ by its center.

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)