Normal Subgroups of the Modular Group Which Are Not Congruence Subgroups

A subgroup of V containing a principal congruence subgroup T(«) is said to be a congruence subgroup, and is of level « if « is the least such integer. In a recent article [2] the writer determined all normal subgroups of T of genus 1 (see [l] 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 T. We let x stand for the substitution