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