Our main purpose is to explore the relationship between
normal subgroups of
Hecke groups and regular maps on compact orientable surfaces. We use regular
maps to find all the normal subgroups of Hecke groups of index
[les ]60. In Sections 1–4 we review the basic results concerning
Hecke groups and normal subgroups of
Fuchsian groups and in Section 5 we outline the basic facts we need
about regular maps. The regular maps of genus 0 are the Platonic solids
and in Section 6 we use
these to completely determine the genus zero normal subgroups
of Hecke groups. The
regular maps of genus 1 were determined this century by Brahana
[1]. We use them
in Sections 7 and 8 to determine the genus 1 normal subgroups
of Hecke groups. We
give alternative proofs and extend theorems of M. Newman and also
Kern-Isberner and Rosenberger. Unlike the genus 0 and 1 cases there are
only finitely many regular
maps of genus g[ges ]2. In Section 9 we use more recent
results concerning their
classification to find all normal subgroups of Hecke groups of
index [les ]60. This was done for the modular group in
[13].