New York Journal of Mathematics
Volume 11 (2005) 205-224


T. W. Müller and J.-C. Schlage-Puchta

Divisibility properties of subgroup numbers for the modular group

Published: June 14, 2005
Keywords: modular group, subgroup numbers, congruences
Subject: 20E

Let Γ=PSL2(Z) be the classical modular group. It has been shown by Stothers (Proc. Royal Soc. Edinburgh 78A, 105-112) that sn, the number of index n subgroups in Γ, is odd if and only if n+3 or n+6 is a 2-power. Moreover, Stothers (loc. cit.) also showed that fλ, the number of free subgroups of index 6λ in Γ, is odd if and only if λ+1 is a 2-power. Here, these divisibility results for fλ and sn are generalized to congruences modulo higher powers of 2. We also determine the behaviour modulo 3 of fλ. Our results are naturally expressed in terms of the binary respectively ternary expansion of the index.

Author information

T. W. Müller:
School of Mathematical Sciences, Queen Mary & Westfield College, University of London, Mile End Road, E1 4NS London, UK

J.-C. Schlage-Puchta:
Universität Freiburg, Mathematisches Institut, Eckerstr. 1, 79104 Freiburg, Germany