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; ω 2 =φ 3 = 1〉. The modular group may also be defined as the group of fractional linear transformations w = ( az+b )/( cz+d ), where a, b, c, d are rational integers with ad − bc = 1. Here, a maximal nonparabolic subgroup of Г is a subgroup that contains no parabolic elements and any proper subgroup of Г which contains S contains 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.