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 Similar groups Explanation
order infinite (countable) The group is infinite because, for instance, it contains all matrices of the form $\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}$ for $a \in \mathbb{Z}$.
As a set, the group is contained in the set of all $2 \times 2$ matrices over $\mathbb{Z}$. This can be identified with $\mathbb{Z}^4$, which is countable since $\mathbb{Z}$ is countable. Thus, $SL(2,\mathbb{Z})$ is also countable.
exponent infinite The group contains the element $\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}$, which has infinite order.
minimum size of generating set 2 Follows from elementary matrices of the first kind generate the special linear group over a Euclidean ring, so $SL(2,\mathbb{Z})$ is generated by all matrices of the form $\begin{pmatrix} 1 & a \\ 0 & 1 \\\end{pmatrix}$ and $pmatrix$ with $a,b$ varying over $\mathbb{Z}$. By the fact that the additive group of $\mathbb{Z}$ is cyclic, all these matrices are generated by $\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}$ and $\begin{pmatrix} 1 & 0 \\ 1 & 1 \\\end{pmatrix}$.
subgroup rank infinite (countable) $SL(2,\mathbb{Z})$ has a subgroup that is isomorphic to free group:F2 (see Sanov subgroup in SL(2,Z) is free of rank two). This in turn has free subgroups of countable rank.

GAP implementation

Description Functions used
SL(2,Integers) SL, Integers