General linear group:GL(2,Z)

Definition
The group $$GL(2,\mathbb{Z})$$ is defined as the group of invertible $$2 \times 2$$ matrices over the ring of integers, under matrix multiplication. Since the determinant is multiplicative and the only invertible integers are $$\pm 1$$, this can equivalently be defined as:

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

This is a particular case of a member of family::general linear group over integers, which in turn is a particular case of a member of family::general linear group.

The subgroup of matrices of determinant one is special linear group:SL(2,Z), and it is a subgroup of index two.

GAP implementation
The group can be defined using GAP's GeneralLinearGroup function, as:

GL(2,Integers)