Commutator subgroups of the extended Hecke groups
Recep Şahin; Osman Bizim; I. N. Cangul
Czechoslovak Mathematical Journal (2004)
- Volume: 54, Issue: 1, page 253-259
- ISSN: 0011-4642
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topŞahin, Recep, Bizim, Osman, and Cangul, I. N.. "Commutator subgroups of the extended Hecke groups $\bar{H}(\lambda _q)$." Czechoslovak Mathematical Journal 54.1 (2004): 253-259. <http://eudml.org/doc/30855>.
@article{Şahin2004,
abstract = {Hecke groups $H(\lambda _q)$ are the discrete subgroups of $\{\mathrm \{P\}SL\}(2,\mathbb \{R\})$ 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.},
author = {Şahin, Recep, Bizim, Osman, Cangul, I. N.},
journal = {Czechoslovak Mathematical Journal},
keywords = {Hecke group; extended Hecke group; commutator subgroup; Hecke group; extended Hecke group; commutator subgroup},
language = {eng},
number = {1},
pages = {253-259},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Commutator subgroups of the extended Hecke groups $\bar\{H\}(\lambda _q)$},
url = {http://eudml.org/doc/30855},
volume = {54},
year = {2004},
}
TY - JOUR
AU - Şahin, Recep
AU - Bizim, Osman
AU - Cangul, I. N.
TI - Commutator subgroups of the extended Hecke groups $\bar{H}(\lambda _q)$
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 1
SP - 253
EP - 259
AB - Hecke groups $H(\lambda _q)$ are the discrete subgroups of ${\mathrm {P}SL}(2,\mathbb {R})$ 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.
LA - eng
KW - Hecke group; extended Hecke group; commutator subgroup; Hecke group; extended Hecke group; commutator subgroup
UR - http://eudml.org/doc/30855
ER -
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