Wohlfahrt's Theorem for the Hecke group G_5

TL;DR

This paper extends Wohlfahrt's theorem to the inhomogeneous Hecke group G_5, establishing a criterion for congruence subgroups based on geometric level and principal congruence subgroups.

Contribution

It generalizes Wohlfahrt's theorem to G_5, providing a new characterization of congruence subgroups in this setting.

Findings

K is congruence iff it contains the principal congruence subgroup of level 2r

The geometric level r determines the congruence property of K

Provides a necessary and sufficient condition for congruence in G_5

Abstract

Let K be a subgroup of the inhomogeneous Hecke group G_5 of finite index. Suppose that the geometric level of K is r. Then K is congruence if and only if K contains the principal congruence subgroup of level 2r.