Arc-transitive digraphs with quasiprimitive local actions

Abstract

Let $\Gamma$ be a finite $G$-vertex-transitive digraph. The in-local action of $(\Gamma ,G)$ is the permutation group $L_{-}$ induced by the vertex-stabiliser on the set of in-neighbours of $v$. The out-local action $L_{+}$ 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.