General linear group:GL(2,Z)

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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 general linear group over integers, which in turn is a particular case of a 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.

Arithmetic functions

Function Value Explanation
order Infinite (countable)
exponent Infinite (countable)
derived length not defined
Frattini length not defined Has a free non-abelian subgroup, so not solvable.

Group properties

Property Satisfied Explanation Comment
abelian group No
nilpotent group No
solvable group No
perfect group No

GAP implementation

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

GL(2,Integers)