Normal congruence subgroups of the Hecke groups G(2^(1∕2)) and G(3^(1∕2))

L. Alayne Parson

Vol. 70 (1977), No. 2, 481–487

DOI: 10.2140/pjm.1977.70.481

Abstract

NormaI congruence subgroups of the classical modular group have been completely classifled by M. Newman and D. McQuillan. In this note we begin the classification of normal congruence subgroups of the Hecke groups G(2^(1∕2)) and G(3^(1∕2)).

Our main result is that if G is a normal congruence subgroup of G(m^(1∕2)), m = 2,3, containing the principal congruence subgroup Γ_m(nm^(1∕2)) where (n,6) = 1 and if G contains only even elements, then G is Γ_m(dm^(1∕2)), Γ_m(dm^(1∕2)) where d∣n or Γ_m(d), Γ_m(d) where d∣n and d > 1. To obtain this result we use facts about the level of a congruence subgroup which are of independent interest.

Mathematical Subject Classification 2000

Primary: 10D05, 10D05

Secondary: 20H10

Milestones

Received: 22 December 1976

Revised: 28 April 1977

Published: 1 June 1977