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 $\mathrm{\Delta }$ where the permutation group $G_x^{\mathrm{\Delta }(x)}$ induced by the stabiliser $G_x$ of the vertex $x$ on the neighbourhood $\mathrm{\Delta }(x)$ satisfies the two conditions given in the introduction. We show that for such a $G$-arc-transitive graph $\mathrm{\Delta }$, if $(x,y)$ is an arc of $\mathrm{\Delta }$, then the subgroup $G_{x,y}^{[1]}$ of $G$ fixing pointwise $\mathrm{\Delta }(x)$ and $\mathrm{\Delta }(y)$ 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.