Arc-transitive digraphs with quasiprimitive local actions

TL;DR

This paper investigates the possible pairs of in- and out-local actions in finite vertex-transitive digraphs, focusing on cases where both actions are quasiprimitive, and provides structural insights into related quasiprimitive groups.

Contribution

It characterizes possible local action pairs in vertex-transitive digraphs, especially when both are quasiprimitive, and establishes a structural result about quasiprimitive group pairs.

Findings

Identifies conditions for pairs of local actions in vertex-transitive digraphs.

Provides structural results on pairs of quasiprimitive groups of the same degree.

Analyzes the case when one quasiprimitive group is a proper quotient of another.

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.