Modular group explained

In mathematics, the modular group is the projective special linear group $\backslash operatorname(2,\backslash mathbb\; Z)$ of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic.

Definition

The modular group is the group of linear fractional transformations of the upper half of the complex plane, which have the form

z\mapsto (az + b)/(cz + d)

where,,, are integers, and . The group operation is function composition.