Commutator Subgroups of the Extended Hecke Groups $$\bar H(\lambda _q )$$

Şahіn, Recep; Bizim, Osman; Cangül, İsmaıl Nacı

doi:10.1023/b:cmaj.0000027265.81403.8d

Cited by 19 publications

(15 citation statements)

References 5 publications

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“…The extended modular group, denoted by H (λ 3 ) = Π = PGL(2, Z), has been 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 [12,13], and [14] 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 (1.1) and Hecke group H (λ q ) is a subgroup of index 2 in H (λ q ).…”

Section: Introductionmentioning

confidence: 99%

Commutator subgroups of the power subgroups of some Hecke groups

Şahіn

Koruoğlu

2010

Ramanujan J

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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.

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Section: Introductionmentioning

confidence: 99%

Commutator subgroups of the power subgroups of some Hecke groups

Şahіn

Koruoğlu

2010

Ramanujan J

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“…2) If p = 2, then the first commutator subgroups H 2,q coincide with the ones given in [16] ii) If p is an odd number and q is an even number, then H p,q is a free group of rank (q − 1)p 2 − pq + 1.…”

Section: Corollary 2 1)mentioning

confidence: 93%

“…Now we focus on the commutator subgroups of H p,q and H p,q . The commutator subgroups of H q , H q , H m q (m−th power subgroup of H q ) and H p,q (p and q are relatively prime) have been studied by many authors see [1], [16], [18], [19], [20] and [22].…”

Section: S(z)mentioning

confidence: 99%

Commutator Subgroups of Generalized Hecke and Extended Generalized Hecke Groups

Kaymak

Demır

Koruoğlu

et al. 2018

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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Let p and q be integers such that 2 ≤ p ≤ q; p + q > 4 and let Hp,q be the generalized Hecke group associated to p and q: The generalized Hecke group Hp,q is generated by X(z) = -(z-λp)-1 and Y (z) = -(z+ λq)-1 where λp = 2 cos ≤ π/p and λq = 2 cos π/q . The extended generalized Hecke group H̅p,q is obtained by adding the reection R(z) = 1/z̅ to the generators of generalized Hecke group Hp,q: In this paper, we study the commutator subgroups of generalized Hecke groups Hp,q and extended generalized Hecke groups H̅p,q.

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“…Also H. q / has the signature .0I 2; q; 1/, that is, all the groups H. q / are triangle groups. The first several of these groups are H. 3 [9,10] and [6]. Thus, the extended Hecke group H .…”

Section: Introductionmentioning

confidence: 99%

“…q / some special cases of q: To do these, we need the following results about the commutator subgroups of H . q / in [9] and [10] .…”

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confidence: 99%