We use regular map theory to obtain all normal subgroups of Hecke groups of genus 0 and 1. The existence of a regular map corresponding uniquely to every normal subgroup of Hecke groups H(\lambda_q) is a result of Jones and Singerman, and it is frequently used here to obtain normal subgroups. It is found that when q is even, H(\lambda_q) has infinitely many normal subgroups on the sphere, while for odd q, this number is finite. The total number of normal subgroups of H(\lambda_q) on a torus is found to be either 0 or infinite. The latter case appears iff q is a multiple of 4. Finally, a result of Rosenberger and Kern-Isberner is reproved here.
Hecke groups, genus, regular maps
CANGÜL, İSMAİL NACİ and BİZİM, OSMAN (1998) "Normal Subgroups of hecke Groups on Sphere and," Turkish Journal of Mathematics: Vol. 22: No. 4, Article 2. Available at: https://journals.tubitak.gov.tr/math/vol22/iss4/2