THE NORMALIZER OF r o (7V) IN PSL(2, R) by M. AKBAS and D. SINGERMAN (Received 17 April, 1989) 1. The structure of the normalizer. Let T denote the modular group, consisting of the Mobius transformations az + b z-> a,b,c,deZ, ad-bc = l. (1) cz + a As usual we denote the above transformation by the matrix V = ( 1 remembering that V and — V represent the same transformation. If N is a positive integer we let T 0 (N) denote the transformations for which c = 0 mod N. Then F 0 (N) is a subgroup of index the product being taken over all prime divisors of N. In this paper we are interested in the normalizer of T 0 (N) in the group PSL(2, U) of all Mobius transformations with real coefficients and determinant one. This normalizer has acquired significance because it is related to the Monster simple group [2]. It has also played an important role in work on Weierstrass points on the Riemann surfaces associated to r o (N), [5], on Modular forms [1] and on Ternary quadratic forms [6]. We denote the normalizer by T B (N) and define B(N) = r fl (A0/r o (A0. Our main result gives the structure of B(N) for all integers N>2. Such a result was given without proof in [1] but we have found several errors in their list, so it may be worthwhile to give a careful treatment. We use the description of the normalizer given by Con way and Norton [2]. No proof was given though a verification can be obtained by the accounts in [1], [5], [7]. The normalizer is given by the transformations corresponding to the matrices '" blh c — de h where all symbols represent integers, h is the largest divisor of 24 such that h 2 | N, e > 0 is an exact divisor of N/h 2 and the determinant of the matrix is e. (We say that e is an exact divisor of M if e \ M and (e, M/e) = 1.) If M =p"'p" 2 • • • p"' is the prime-power decomposition of M then M has T exact divisors, all of the form p?'p£ 2 ... p p r ' where j8, = 0 or or, for i = 1, 2,..., r. We denote the set of exact divisors of M by Ex(Af). Our investigation into B(N) and r B (N) is facilitated by the observation that Ex(M) is a group with respect to a suitable binary operation. Glasgow Math. J. 32 (1990) 317-327. https://www.cambridge.org/core/terms. https://doi.org/10.1017/S001708950000940X Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 02 Jun 2020 at 00:42:46, subject to the Cambridge Core terms of use, available at