Arc-transitive digraphs with quasiprimitive local actions
Michael Giudici, S. P. Glasby, Cai Heng Li, Gabriel Verret

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 be a finite -vertex-transitive digraph. The in-local action of is the permutation group induced by the vertex-stabiliser on the set of in-neighbours of . The out-local action is defined analogously. Note that and may not be isomorphic. We thus consider the problem of determining which pairs are possible. We prove some general results, but pay special attention to the case when and 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.
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Taxonomy
TopicsFinite Group Theory Research · graph theory and CDMA systems · Coding theory and cryptography
