Nonparabolic subgroups of the modular group

Abstract

In this paper we shall discuss maximal nonparabolic and maximal normal nonparabolic subgroups of the modular group Г = 〈ω, φ; ω2=φ3= 1〉. The modular group may also be defined as the group of fractional linear transformationsw= (az+b)/(cz+d), wherea, b, c, dare rational integers withad − bc= 1. Here, a maximal nonparabolic subgroup of Г is a subgroup that contains no parabolic elements and any proper subgroup of Г which containsScontains parabolic elements. Similarly, a maximal normal nonparabolic subgroup is a normal nonparabolic subgroup of Г which is not contained in any larger normal nonparabolic subgroup of Г.