Special linear group:SL(2,Z)

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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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)