On the extenden modular group

In this study, some normal subgroups of the extended modular group are considered. The thesis consists of five chapters. In the second chapter of the thesis, the definitions to be used in the study are given. In addition, the necessary theorems and methods are presented.. In the third chapter of the thesis, the power, commutator, commutator of power, and principal congruence subgroups of the modular group are examined. The indices of these subgroups in the modular group were found, and their generators were found using the Reidemeister-Schreier method. In addition, the symbols of these subgroups were obtained by the permutation method, and their genera were obtained by the Riemann-Hurwitz formula. Finally, the existing relationships between the studied subgroups were found. In the fourth chapter of the thesis, the generators and group presentations of the commutator and power subgroups of the extended modular group are given. In addition, some results are given about the principal congruence subgroups of the extended modular group. In the fifth and last chapter of the thesis, the results are summarized and suggestions for future studies are given.