Coset Diagrams and Relations for PSL (2,Z) PDF

Keywords

Mathematics, vertices, coset diagrams

Abstract

A diagramatic argument, called coset diagrams for the modular group PSL(2,Z), is used to prove the results stated in this paper. Let G denote the subgroup of the modular group PSL(2,Z), generated by the linear-fractional transformations x and y where x and y are respectively defined as z-+ -li z and z-+ (z -1)/ z. A diagram with n vertices depicts a (transitive) permutation representation of the modular group: fixed points of x and y are defined by heavy dots, and 3-cycles of y by triangles whose vertices are permuted anti-clockwise by y; and any two vertices which are interchanged by x are joined by an edge. In this paper we have shown that the coset diagram for the action of G on the rational projective line is connected, and transitive. Using these coset diagrams we have shown that the group PSL(2,Z) is generated by the Iinear2 fractional transformations x and y and that x= i = I are defming relations for PSL(2,Z).