Elliptic Elements of a Subgroup of the Normalizer and Circuits in Orbital Graphs

In this study, we investigate suborbital graphs G_{u, N} of the normalizer Gamma_B(N) of Gamma_0(N) in PSL(2, R) for N = 2^{alpha} 3^{beta} \u3e 1 where alpha = 0, 2, 4, 6, and beta = 0, 2. In these cases the normalizer becomes a triangular group. We first define an imprimitive action of Gamma_B (N) on ^Q using the group Gamma^0_C (N) and then obtain some properties of the suborbital graphs arising from this action. Finally we define suborbital graphs F_{u;N} and investigate their properties. As a consequence, we find some certain relationships between the lengths of circuits in suborbital graphs F_{u;N} and the periods of the group Gamma^0_C (N)