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Concept
Modular group
Summary
In mathematics, the modular group is the projective special linear group of 2 × 2 matrices with integer coefficients and determinant 1. The matrices A and −A 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.
The modular group Γ is the group of linear fractional transformations of the upper half of the complex plane, which have the form
where a, b, c, d are integers, and ad − bc = 1. The group operation is function composition.
This group of transformations is isomorphic to the projective special linear group PSL(2, Z), which is the quotient of the 2-dimensional special linear group SL(2, Z) over the integers by its center {I, −I}. In other words, PSL(2, Z) consists of all matrices
where a, b, c, d are integers, ad − bc = 1, and pairs of matrices A and −A are considered to be identical. The group operation is the usual multiplication of matrices.
Some authors define the modular group to be PSL(2, Z), and still others define the modular group to be the larger group SL(2, Z).
Some mathematical relations require the consideration of the group GL(2, Z) of matrices with determinant plus or minus one. (SL(2, Z) is a subgroup of this group.) Similarly, PGL(2, Z) is the quotient group GL(2, Z)/{I, −I}. A 2 × 2 matrix with unit determinant is a symplectic matrix, and thus SL(2, Z) = Sp(2, Z), the symplectic group of 2 × 2 matrices.
To find an explicit matrix in SL(2, Z), begin with two coprime integers , and solve the determinant equation(Notice the determinant equation forces to be coprime since otherwise there would be a factor such that , , hencewould have no integer solutions.) For example, if then the determinant equation readsthen taking and gives , henceis a matrix. Then, using the projection, these matrices define elements in PSL(2, Z).
The unit determinant of
implies that the fractions a/b, a/c, c/d, b/d are all irreducible, that is having no common factors (provided the denominators are non-zero, of course).
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