Modular group

In mathematics, the Modular group Γ or PSL(2,Z) is the 2-dimensional projective special linear group over the integers. In other words, the modular group consists of all matrices

${\begin{bmatrix}a&b\\c&d\end{bmatrix}}$

where a, b, c, and d are integers, ad - bc = 1, and we identify two matrices if their corresponding entries each differ by a factor of -1. The operation is the usual multiplication of matrices.

The modular group is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. The modular group is important because it forms a subgroup of the group of isometries of the hyperbolic plane. If we consider the upper half-plane model of hyperbolic plane geometry, the isometry group is given by all Möbius transformations of the form

$w={\frac {az+b}{cz+d}}$

An alternative representation of Γ is the group of all (2x2)-matrices with determinant 1; it can then be shown that every A ∈ Γ can be non-uniquely written as

$A=T^{n_{1}}ST^{n_{2}}S\dots ST^{n_{k}}$

for some k > 0 and coefficients n_i, where

$T={\begin{bmatrix}0&-1\\1&0\end{bmatrix}}$

and

$S={\begin{bmatrix}1&1\\0&1\end{bmatrix}}$

External links

• MathWorld: Modular group Γ