|
In mathematics, the modular group is the projective special linear group ''PSL''(2,Z) of 2 x 2 matrices with integer coefficients and unit determinant. 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. ==Definition== 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'', and ''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 . In other words, PSL(2, Z) consists of all matrices : where ''a'', ''b'', ''c'', and ''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)/. A 2 × 2 matrix with unit determinant is a symplectic matrix, and thus SL(2, Z) = Sp(2, Z), the symplectic group of 2x2 matrices. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Modular group」の詳細全文を読む スポンサード リンク
|