On the structure of vertex stabilizers of arc-transitive locally quasiprimitive graphs

Let Delta be a connected arc-transitive G-graph which is locally finite and locally quasiprimitive. Let {x, y} be an edge of Delta. A relation between G(x)([1])/O-p(G(x)([1]))and the existence of certain normal subgroups of G(x )(Delta(x))and G(x,y )(Delta(x))is established. This is then used to determine the vertex stabilizers of a class of 2-arc-transitive graphs with trivial edge kernel