Classification of Normal Subgroups of the Modular Group

where a, b, c, d are rational integers and ad-bc= 1. Then T is generated by the linear fractional transformations S, T where St = t+ 1, 7V= — 1/t and is the free product of the cyclic group {T} of order 2 and the cyclic group {ST} of order 3. Let G be a normal subgroup of T of finite index p. It is known that there is just one such subgroup for p=l, 2, 3 which may be described as T", the subgroup of r generated by the /¿th powers of the elements of T. In all other cases p. must be a multiple of 6 (see [3] or [6]) and there are only finitely many normal subgroups of r of a given finite index p, since the total number of subgroups of a given finite index in a finitely generated group is finite. The purpose of this article is to obtain some information about the function N{p), the number of normal subgroups of T of index p. By the remarks above N{l) = N{2)=N{3) = 1, and N{p)=0 if p> 3 and p^O (mod 6). We will determine all normal subgroups of T for p. = 66, and we will also determine N{p) explicitly when p=6q or I2q, where a is a prime. For example it will be shown that N{6q) = 1 +{q/3) for all primes q>3, and that iV(12a) = 0 for all primes q> 11. Here (a/3) is the Legendre-Jacobi symbol of quadratic reciprocity. Some recent work of I. M. S. Dey [2] implies that the total number M{p) of subgroups of T of index p. satisfies the recurrence formula