Divisibility properties of subgroup numbers for the modular group
Thomas Müller, Jan‐Christoph Schlage‐Puchta · Ghent University Academic Bibliography (Ghent University) · 2005
Let Γ = PSL 2 (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 f λ , 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 f λ and sn are generalized to congruences modulo higher powers of 2. We also determine the behaviour modulo 3 of f λ .Our results are naturally expressed in terms of the binary respectively ternary expansion of the index.Contents 1. Introduction and results 205 2. Proof of Theorem 1 208 3. Proof of Proposition 1 213 4. Proof of Theorem 2 216 5. Proof of Theorem 3 221 References 223