The subgroups of modular groups

IV ABSTRACT In this thesis after modular groups was mentioned the subgroups of modular groups are studied. In chapter I firstly homogeneous and inhomogeneous lineer transformation concepts and relation bnetween these two concepts are considered. The modular group is defined by using this transformations. After this modular transformations are clasified by means of the transformations and also generators of the modular groups and their relations are studied. More over, fundemental region is defined. Teselletion upper half plane is examined because of existence of fundemental region. In chapter II the subgroup of modular groups are studied with details. Principle congruence groups and special congruence groups have been defined. After that principle region was shown for subgroups. The quotiet space Jf/T is studied in details and the genous formula of principal regions is given. The branch schemas corresponding to normal subgroups was drawn. Finally the genous of To(N) of principal region has been studied.