Editing Modular group (section)

==Definition==
The '''modular group''' {{math|Γ}} is the [[Group (mathematics)|group]] of [[Möbius transformation|fractional linear transformation]]s of the [[Upper half-plane#Complex plane|complex upper half-plane]], which have the form

:<math>z\mapsto\frac{az+b}{cz+d},</math>

where <math>a,b,c,d</math> are integers, and <math>ad-bc=1</math>. The group operation is [[function composition]].

This group of transformations is isomorphic to the [[projective linear group|projective special linear group]] <math> \operatorname{PSL}(2,\mathbb Z)</math>, which is the quotient of the 2-dimensional [[special linear group]] <math> \operatorname{SL}(2,\mathbb Z)</math> by its [[center (group theory)|center]] <math>\{I,-I\}</math>.  In other words, <math> \operatorname{PSL}(2,\mathbb Z)</math> consists of all matrices

:<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix}</math>

where <math>a,b,c,d</math> are integers, <math>ad-bc=1</math>, and pairs of matrices <math>A</math> and <math>-A</math> are considered to be identical.  The group operation is usual [[matrix multiplication]].

Some authors ''define'' the modular group to be <math> \operatorname{PSL}(2,\mathbb Z)</math>, and still others define the modular group to be the larger group <math> \operatorname{SL}(2,\mathbb Z)</math>. 

Some mathematical relations require the consideration of the group <math>\operatorname{GL}(2,\mathbb Z)</math> of matrices with determinant plus or minus one. (<math>\operatorname{SL}(2,\mathbb Z)</math> is a subgroup of this group.)  Similarly, <math>\operatorname{PGL}(2,\mathbb Z)</math> is the quotient group <math>\operatorname{GL}(2,\mathbb Z)/\{I,-I\}</math>. 

Since all <math>2\times 2</math> matrices with determinant 1 are [[symplectic matrix|symplectic matrices]], then <math>\operatorname{SL}(2,\mathbb Z)=\operatorname{Sp}(2,\Z)</math>, the [[symplectic group]] of <math>2\times 2</math> matrices.

=== Finding elements ===
To find an explicit matrix
:<math>\begin{pmatrix} a & x \\ b & y \end{pmatrix}</math>
in <math>\operatorname{SL}(2,\mathbb Z)</math>, begin with two coprime integers <math>a,b</math>, and solve the determinant equation <math>ay-bx = 1</math>.{{efn|Notice the determinant equation forces <math>a,b</math> to be coprime, since otherwise there would be a factor <math>c>1</math> such that <math>ca' = a</math> and <math>cb' = b</math>, hence <math>c(a'y-b'x) = 1</math> would have no integer solutions.}} 

For example, if <math>a = 7, \text{ } b =6 </math> then the determinant equation reads
:<math>7y-6x = 1,</math>
then taking <math>y = -5</math> and <math>x = -6</math> gives <math>-35 - (-36) = 1</math>. Hence
:<math>\begin{pmatrix}
7 & -6 \\
6 & -5
\end{pmatrix}</math>
is a matrix. Then, using the projection, these matrices define elements in <math>\operatorname{PSL}(2,\mathbb Z)</math>.