In this paper we study G-arc-transitive graphs Ξ where the
permutation group GxΞ(x)β induced by the stabiliser Gxβ of the
vertex x on the neighbourhood Ξ(x) satisfies the two conditions given
in the introduction. We show that for such a G-arc-transitive graph Ξ,
if (x,y) is an arc of Ξ, then the subgroup Gx,y[1]β of G
fixing pointwise Ξ(x) and Ξ(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 Gx,y[1]β has arbitrarily
large composition factors