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 D where the permutation group G D x induced by the stabiliser G x of the vertex x on the neighbourhood D(x) satisfies the two conditions given in the introduction. We show that for such a G-arc-transitive graph D, if x; is an arc of D, then the subgroup G 1x; y of G fixing D and D point-wise 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 1x; y has arbitrarily large composition factors. © Gruyter 2012.