Two local conditions on the vertex stabiliser of arc-transitive graphs and their effect on the Sylow subgroups

In this paper we study $G$-arc-transitive graphs $\Delta$ where the permutation group $G_x^{\Delta(x)}$ induced by the stabiliser $G_x$ of the vertex $x$ on the neighbourhood $\Delta(x)$ satisfies the two conditions given in the introduction. We show that for such a $G$-arc-transitive graph $\Delta$, if $(x,y)$ is an arc of $\Delta$, then the subgroup $G_{x,y}^{[1]}$ of $G$ fixing pointwise $\Delta(x)$ and $\Delta(y)$ is a $p$-group for some prime $p$. Next we prove that every $G$-locally primitive (respectively quasiprimitive, semiprimitive) graph satisfies our two local hypotheses. Thus this provides a new Thompson-Wielandt-like theorem for a very large class of arc-transitive graphs. Furthermore, we give various families of $G$-arc-transitive graphs where our two local conditions do not apply and where $G_{x,y}^{[1]}$ has arbitrarily large composition factors

On congruence equations arising from suborbital graphs

In this paper we deal with congruence equations arising from suborbital graphs of the normalizer of Γ_0(m) in PSL(2,R) . We also propose a conjecture concerning the suborbital graphs of the normalizer and the related congruence equations. In order to prove the existence of solution of an equation over prime finite field, this paper utilizes the Fuchsian group action on the upper half plane and Farey graphs properties

Some remarks on orbital digraphs for the finite primitive groups

In this paper we concern with the relationship between the finite groups PSL(2, q), q>5 a prime, and orbital digraphs. And also we explain that for a generator elliptic element in permutation group, there is a hyperbolic circuit in suborbital graph

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)