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

Abstract

AbstractLet $$\Delta $$ Δ be a connected arc-transitive G-graph which is locally finite and locally quasiprimitive. Let $$\{x,y\}$$ { x , y } be an edge of $$\Delta $$ Δ . A relation between $$G_x^{[1]}/O_p(G_x^{[1]})$$ G x [ 1 ] / O p ( G x [ 1 ] ) and the existence of certain normal subgroups of $$G_x^{\Delta (x)}$$ G x Δ ( x ) and $$G_{x,y}^{\Delta (x)}$$ 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.