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)