关键词: Extended Hecke group, even subgroup, commutator subgroup, principal congruence
subgroup

Abstract:

The authors consider the extended Hecke groups $\overline{H}$($\lambda _q)$ generated
by $T(z)=-1$ $/$ $z$, $S(z)=-1/(z+\lambda _q)$ and $R(z)=1$ $/$ $\overline{z}
$ with $\lambda _q=2\cos (\pi /q)$ for $q\geq 3$ an integer$.$ In this
paper, the even subgroup $\overline{H}_e$($\lambda _q),$ the second
commutator subgroup $\overline{H}^{\prime \prime }$($\lambda _q)$ and the
principal congruence subgroups $\overline{H}_p$($\lambda _q)$ of the
extended Hecke groups $\overline{H}$($\lambda _q)$ are studied. Also,
 relations between them are given.

Key words: Extended Hecke group, even subgroup, commutator subgroup, principal congruence
subgroup