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Bounding $s$ for vertex-primitive $s$-arc-transitive digraphs of alternating and symmetric groups

Junyan Chen, Lei Chen, Michael Giudici, Jing Jian Li, Cheryl E. Praeger, Binzhou Xia

Abstract

Determining an upper bound on $s$ for finite vertex-primitive $s$-arc-transitive digraphs has received considerable attention dating back to a question of Praeger in 1990. It was shown by Giudici and Xia that the smallest upper bound on $s$ is attained for some digraph admitting an almost simple $s$-arc-transitive group. In this paper, based on the work of Pan, Wu and Yin, we prove that $s\leqslant 2$ in the case where the group is an alternating or symmetric group.

Bounding $s$ for vertex-primitive $s$-arc-transitive digraphs of alternating and symmetric groups

Abstract

Determining an upper bound on for finite vertex-primitive -arc-transitive digraphs has received considerable attention dating back to a question of Praeger in 1990. It was shown by Giudici and Xia that the smallest upper bound on is attained for some digraph admitting an almost simple -arc-transitive group. In this paper, based on the work of Pan, Wu and Yin, we prove that in the case where the group is an alternating or symmetric group.

Paper Structure

This paper contains 5 sections, 20 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. On maximal subgroups of alternating groups
  4. Homogeneous factorisations of wreath product
  5. Proof of the main theorem

Key Result

Theorem 1.2

Let $\Gamma$ be a $G$-vertex-primitive $(G,s)$-arc-transitive digraph where $G$ is almost simple with socle $A_n$, and let $v$ be a vertex of $\Gamma$. Then one of the following holds:

Theorems & Definitions (32)

  • Theorem 1.2: Pan-Wu-Yin
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4: O'Nan-Scott
  • Lemma 2.5
  • Theorem 2.6: Liebeck-Praeger-Saxl
  • Lemma 2.7
  • ...and 22 more