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

Abstract Let Δ be a connected arc-transitive G -graph which is locally finite and locally quasiprimitive. Let \x, y\ x, y be an edge of 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 Gₓ^ (x) G x Δ (x) and Gₗ, ₘ^ (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.