Commutator subgroups of the power subgroups of some Hecke groups

Abstract

Let q ≥ 3 be a prime and let H (λ q ) be the Hecke group associated to q. Let m be a positive integer and H m (λ q ) be the mth power subgroup of H (λ q ). In this work, we study the commutator subgroups of the power subgroups H m (λ q ) of H (λ q ). Then, we give the derived series for all triangle groups of the form (0; 2, q, n) for n a positive integer, since there is a nice connection between the signatures of the subgroups we studied and the signatures of these derived series.

Commutator subgroups of the power subgroups of Hecke groups H(λq) II

“…In Theorem 2.2, if d = q then q d − 1 = 0.Then there is no d given in the last paragraph of the proof of ii). Thus there are onlyd 2 + 2 × d 3 + · · · + (d − 1) × d d = 1 + (d − 2)2 d−1 generators of (H m ) (λ q ).In this case, this result coincides with the Theorem 2.2 in[6].…”

“…In the above cases i), ii), iii * ) and iv * ), then all results for q 3 an odd number coincide with the ones for q 3 a prime number in [6]. In the case q 3 an odd number, there are only two cases different from q 3 prime number case in [6]. These are the cases iii) and iv) except for the cases iii * ) and iv * ).…”

“…In this note, we generalize the results given in [6] for q 3 a prime number to q 3 an odd number. In fact, there are similar cases between the case of prime q and the case of odd q.…”

“…In [6], according to all the cases m positive integer, Sahin and Koruoglu obtained the group structures and the signatures of commutator subgroups of the power subgroups H m (λ q ) of the Hecke groups H(λ q ), q 3 a prime.…”

Some results on Hecke and extended Hecke groups

“…Hence the surface The extended modular group, denoted by H (λ 3 ) = Π = P GL (2, Z), is defined by adding the reflection R(z) = 1/ z to the generators of the modular group H (λ 3 ) . Then the extended Hecke group, denoted by H(λ q ), has been defined in [35] and [39] similar to the extended modular group by adding the reflection R(z) = 1/z to the generators of the Hecke group H(λ q ). Thus the extended Hecke group H(λ q ) has the presentation…”