s-Arc-transitive graphs and normal subgroups

We study s  -arc-transitive graphs with s⩾2s⩾2, and give a characterisation of the actions of vertex-transitive normal subgroups. An interesting consequence of this characterisation states that each non-bipartite 3-arc-transitive graph is a normal cover of a 2-arc-transitive graph admitting a simple group, or a locally primitive graph admitting a simple group with a soluble vertex stabiliser.