Abstract
In this paper, first, we determine the quotient groups of the Hecke groups H(lambda(q)), where q >= 7 is prime, by their principal congruence subgroups H-p(lambda(q)) of level p, where p is also prime. We deal with the case of q = 7 separately, because of its close relation with the Hurwitz groups. Then, using the obtained results, we find the principal congruence subgroups of the extended Hecke groups (H) over tilde(lambda(q)) for q >= 5 prime. Finally, we show that some of the quotient groups of the Hecke group H(lambda(q)) and the extended Hecke group (H) over bar (lambda(q)), q >= 5 prime, by their principal congruence subgroups H-p(lambda(q)) are M*-groups.