Normal subgroups of the modular group which are not congruence subgroups

Abstract

A subgroup of r containing a principal congruence subgroup r(n) is said to be a congruence subgroup, and is of level n if n is the least such integer. In a recent article [2] the writer determined all normal subgroups of r of genus 1 (see [1 ] for the definition of the genus of a subgroup of r). An interesting question that arises is to decide which of these are also congruence subgroups. In this note we show that there are just 4 such groups. This furnishes a new family of normal subgroups of finite index in the modular group which are not congruence groups (see [3 ] for other examples). The proof makes use of some recent work of K. Wohlfahrt [4] on the definition of level for an arbitrary subgroup of finite index in r. We let x stand for the substitution