---
tag: VEKV1EM77B
kind: object
name: Modular group
msc: [11F06, 20H05]
instance_of: [G5S1M9]
uses: [0V9PMN0RBH, 0SG74WBFA2, X6XGGH5FPW, 0KYX2N69J0]
sources:
  - url: https://en.wikipedia.org/wiki/Modular_group
same_as:
  wikidata: Q1834342
  wikipedia: Modular group
---

## Description

The modular group is the [[special linear group|0V9PMN0RBH]]
$$\mathrm{SL}_2(\Z) = \set{\begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, d \in \Z, \ ad - bc = 1}$$
under matrix multiplication.
It acts on the [[upper half-plane|X6XGGH5FPW]] $\mathbb{H}$ by [[Möbius transformations|0SG74WBFA2]] $\tau \mapsto (a\tau + b)/(c\tau + d)$.
The matrices $I$ and $-I$ both act as the identity map of $\mathbb{H}$, so many authors call the [[quotient group|0KYX2N69J0]] $\mathrm{PSL}_2(\Z) = \mathrm{SL}_2(\Z) / \set{\pm I}$ the modular group.
The group is generated by
$$S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \qquad T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix},$$
which act by $S\tau = -1/\tau$ and $T\tau = \tau + 1$.

## Remarks

The generators satisfy $S^2 = -I$ and $(ST)^3 = -I$.
Check: $ST = \begin{pmatrix} 0 & -1 \\ 1 & 1 \end{pmatrix}$, $(ST)^2 = \begin{pmatrix} -1 & -1 \\ 1 & 0 \end{pmatrix}$, and $(ST)^3 = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$.
The group $\mathrm{PSL}_2(\Z)$ is the free product of a cyclic group of order $2$ and a cyclic group of order $3$.
The set $\set{\tau \in \mathbb{H} : \abs{\tau} \ge 1, \ \abs{\operatorname{Re} \tau} \le 1/2}$ is a fundamental domain for its action on $\mathbb{H}$.
The [[modular forms|GWKN5VG78A]] are the functions on $\mathbb{H}$ with a prescribed behaviour under this group.

## Explanations

### L1 human

The modular group is the group of $2 \times 2$ integer matrices with determinant $1$.
Each such matrix moves the points of the upper half-plane by a fraction $(a\tau + b)/(c\tau + d)$.
Every such move is built from two simple ones: the shift $\tau \mapsto \tau + 1$ and the inversion $\tau \mapsto -1/\tau$.
The group also describes the ways to change an oriented basis of a lattice in the plane.