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Two local conditions on the vertex stabiliser of arc-transitive graphs and their effect on the Sylow subgroups

Abstract

In this paper we study GG-arc-transitive graphs Ξ”\Delta where the permutation group GxΞ”(x)G_x^{\Delta(x)} induced by the stabiliser GxG_x of the vertex xx on the neighbourhood Ξ”(x)\Delta(x) satisfies the two conditions given in the introduction. We show that for such a GG-arc-transitive graph Ξ”\Delta, if (x,y)(x,y) is an arc of Ξ”\Delta, then the subgroup Gx,y[1]G_{x,y}^{[1]} of GG fixing pointwise Ξ”(x)\Delta(x) and Ξ”(y)\Delta(y) is a pp-group for some prime pp. Next we prove that every GG-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 GG-arc-transitive graphs where our two local conditions do not apply and where Gx,y[1]G_{x,y}^{[1]} has arbitrarily large composition factors

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