Vertex-primitive s-arc-transitive digraphs of alternating and symmetric groups

TL;DR

This paper proves that for vertex-primitive s-arc-transitive digraphs with alternating or symmetric groups, the parameter s is at most 2, advancing understanding of symmetry properties in these structures.

Contribution

It establishes an upper bound of s ≤ 2 for such digraphs with alternating or symmetric groups, a significant step in the broader problem of classifying these digraphs.

Findings

s ≤ 2 for all G-vertex-primitive s-arc-transitive digraphs with G an alternating or symmetric group

Methods developed may apply to other almost simple groups cases

Progress towards bounding s in vertex-primitive s-arc-transitive digraphs

Abstract

A fascinating problem on digraphs is the existence problem of the finiteupper bound on s for all vertex-primitive s-arc-transitive digraphs except directed cycles (which is known to be reduced to the almost simple groups case). In this paper,we prove that s 2 for all G-vertex-primitive s-arc-transitive digraphs with G an (insoluble) alternating or symmetric group, which makes an important progress towards a solution of the problem. The proofs involves some methods that may be used to investigate other almost simple groups cases.