Finite symmetric graphs with two-arc transitive quotients

This paper forms part of a study of 2-arc transitivity for finite imprimitive symmetric graphs, namely for graphs Γ admitting an automorphism group G that is transitive on ordered pairs of adjacent vertices, and leaves invariant a nontrivial vertex partition B . Such a group G is also transitive on the ordered pairs of adjacent vertices of the quotient graph Γ B corresponding to B . If in addition G is transitive on the 2-arcs of Γ (that is, on vertex triples ( α , β , γ ) such that α , β and β , γ are adjacent and α ≠ γ ), then G is not in general transitive on the 2-arcs of Γ B , although it does have this property in the special case where B is the orbit set of a vertex-intransitive normal subgroup of G. On the other hand, G is sometimes transitive on the 2-arcs of Γ B even if it is not transitive on the 2-arcs of Γ . We study conditions under which G is transitive on the 2-arcs of Γ B . Our conditions relate to the structure of the bipartite graph induced on B ∪ C for adjacent blocks B , C of B , and a graph structure induced on B. We obtain necessary and sufficient conditions for G to be transitive on the 2-arcs of one or both of Γ , Γ B for certain families of imprimitive symmetric graphs.