  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.
