# Article

 Title: Commutator subgroups of the extended Hecke groups $\bar{H}(\lambda_q)$ (English) Author: Sahin, R. Author: Bizim, O. Author: Cangul, I. N. Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 54 Issue: 1 Year: 2004 Pages: 253-259 Summary lang: English . Category: math . Summary: Hecke groups $H(\lambda _q)$ are the discrete subgroups of ${\mathrm PSL}(2,\mathbb{R})$ generated by $S(z)=-(z+\lambda _q)^{-1}$ and $T(z)=-\frac{1}{z}$. The commutator subgroup of $H$($\lambda _q)$, denoted by $H^{\prime }(\lambda _q)$, is studied in [2]. It was shown that $H^{\prime }(\lambda _q)$ is a free group of rank $q-1$. Here the extended Hecke groups $\bar{H}(\lambda _q)$, obtained by adjoining $R_1(z)=1/\bar{z}$ to the generators of $H(\lambda _q)$, are considered. The commutator subgroup of $\bar{H}(\lambda _q)$ is shown to be a free product of two finite cyclic groups. Also it is interesting to note that while in the $H(\lambda _q)$ case, the index of $H^{\prime }(\lambda _q)$ is changed by $q$, in the case of $\bar{H}(\lambda _q)$, this number is either 4 for $q$ odd or 8 for $q$ even. (English) Keyword: Hecke group Keyword: extended Hecke group Keyword: commutator subgroup MSC: 11F06 MSC: 20H05 MSC: 20H10 idZBL: Zbl 1053.11038 idMR: MR2040237 . Date available: 2009-09-24T11:11:59Z Last updated: 2020-07-03 Stable URL: http://hdl.handle.net/10338.dmlcz/127882 . Reference: [1] R. B. J. T.  Allenby: Rings, Fields and Groups. Second Edition.Edward Arnold, London-New York-Melbourne-Auckland, 1991. MR 1144518 Reference: [2] I. N.  Cangül and D.  Singerman: Normal subgroups of Hecke groups and regular maps.Math. Proc. Camb. Phil. Soc. 123 (1998), 59–74. MR 1474865, 10.1017/S0305004197002004 Reference: [3] H. S. M.  Coxeter and W. O. J. Moser: Generators and Relations for Discrete Groups.Springer, Berlin, 1957. MR 0088489 Reference: [4] E. Hecke: Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichungen.Math. Ann. 112 (1936), 664–699. MR 1513069, 10.1007/BF01565437 Reference: [5] D. L. Johnson: Topics in the Theory of Group Presentations. L.M.S. Lecture Note Series  42.Cambridge Univ. Press, Cambridge, 1980. MR 0695161 Reference: [6] G. A. Jones and J. S.  Thornton: Automorphisms and congruence subgroups of the extended modular group.J.  London Math. Soc. 34 (1986), 26–40. MR 0859146, 10.1112/jlms/s2-34.1.26 .

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