 

T. W. Müller and J.C. SchlagePuchta
Divisibility properties of subgroup numbers for the modular group


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


Abstract
Let Γ=PSL_{2}(Z) be the classical modular group. It
has been shown by Stothers (Proc. Royal Soc. Edinburgh
78A, 105112) that s_{n}, the number of index n subgroups in
Γ, is odd if and only if n+3 or n+6 is a 2power.
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 2power. Here, these
divisibility results for f_{λ} and s_{n} 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.


Author information
T. W. Müller:
School of Mathematical Sciences, Queen Mary & Westfield College, University of London, Mile End Road, E1 4NS London, UK
T.W.Muller@qmul.ac.uk
J.C. SchlagePuchta:
Universität Freiburg, Mathematisches Institut, Eckerstr. 1, 79104 Freiburg, Germany
jcp@math.unifreiburg.de

