Arc-transitive digraphs with quasiprimitive local actions

Abstract

Let Γ be a finite G-vertex-transitive digraph. The in-local action of (Γ,G) is the permutation group L− induced by a vertex-stabiliser on the set of in-neighbours of the corresponding vertex. The out-local actionL+ is defined analogously. Note that L− and L+ may not be isomorphic. We thus consider the problem of determining which pairs (L−,L+) are possible. We prove some general results, but pay special attention to the case when L− and L+ are both quasiprimitive. (Recall that a permutation group is quasiprimitive if each of its nontrivial normal subgroups is transitive.) Along the way, we prove a structural result about pairs of finite quasiprimitive groups of the same degree, one being (abstractly) isomorphic to a proper quotient of the other.