Commutator subgroups of the power subgroups of generalized Hecke groups
Let \(p\), \(q\geq 2\) be relatively prime integers and let \(H_{p,q}\) be the generalized Hecke group associated to \(p\) and \(q\). The generalized Hecke group \(H_{p,q}\) is generated by \(X(z)=-(z-\lambda _{p})^{-1}\) and \(Y(z)=-(z+\lambda_{q})^{-1}\) where \(\lambda _{p}=2\cos \frac{\pi }{p}\)...
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| Date: | 2019 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2019
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/597 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| Summary: | Let \(p\), \(q\geq 2\) be relatively prime integers and let \(H_{p,q}\) be the generalized Hecke group associated to \(p\) and \(q\). The generalized Hecke group \(H_{p,q}\) is generated by \(X(z)=-(z-\lambda _{p})^{-1}\) and \(Y(z)=-(z+\lambda_{q})^{-1}\) where \(\lambda _{p}=2\cos \frac{\pi }{p}\) and \(\lambda_{q}=2\cos \frac{\pi }{q}\). In this paper, for positive integer \(m\), we study the commutator subgroups \((H_{p,q}^{m})'\) of the power subgroups \(H_{p,q}^{m}\) of generalized Hecke groups \(H_{p,q}\). We give an application related with the derived series for all triangle groups of the form \((0;p,q,n)\), for distinct primes \(p\), \(q\) and for positive integer \(n\). |
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