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Note on normal subgroups of the modular group

L. Greenberg
1966 Proceedings of the American Mathematical Society
Let r denote the modular group, namely the group of linear fractional transformations az + b T(z) = -, cz + d where a, b, c, d are integers and ad -be = 1. It is well known that the transformations generate T, with defining relations X2 = Y3 = 1. We shall often need to consider the element Z = XY, which is a parabolic transformation. Any parabolic transformation in V is conjugate to a power of Z. Let A be a normal subgroup of finite index p in T. The level n of N is defined as the least
more » ... s the least positive integer such that ZnEN. The conjugacy class of Z in r splits up into a finite number of equivalence classes under conjugacy by N. The number t of equivalence classes is called the parabolic class number of N. It is known that the integers p, n, t satisfy the relation (1) M = nt. One way of seeing this relation is the following. T operates discontinuously in the upper half-plane D. We obtain quotient surfaces St = D/Y and Sn = D/N. Since A is a normal subgroup of Y, we have a normal (branched) covering Sr-The covering has p sheets and there are exactly t points lying over p, each of which has branching order n -1 (i.e. n sheets meet at each pi). By counting the sheets over p, we find p = nt.