# 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)