On Suborbital Graphs for the Normalizer of $\Gamma_{0}(N)$

Abstract

In this study, we deal with the conjecture given in [R. Keskin, Suborbital graph for the normalizer of $\Gamma _{0}(m)$, European Journal of Combinatorics 27 (2006) 193-206.], that when the normalizer of $\Gamma _{0}(N)$ acts transitively on ${\Bbb Q\cup\{\infty \}}$, any circuit in the suborbital graph $G(\infty,u/n)$ for the normalizer of $\Gamma _{0}(N),$ is of the form $$ v\rightarrow T(v)\rightarrow T^{2}(v)\rightarrow {\ \cdot \cdot \cdot } \rightarrow T^{k-1}(v)\rightarrow v, $$ where $n>1$, $v\in {\Bbb Q\cup \{\infty \}}$ and $T$ is an elliptic mapping of order $k$ in the normalizer of $\Gamma_{0}(N)$.