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

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