Vertex-primitive s-arc-transitive digraphs of linear groups

Abstract

We study G-vertex-primitive and $(G,s)$-arc-transitive digraphs for almost simple groups G with socle ${\mathrm{PSL}}_n(q)$. We prove that $s⩽2$ for such digraphs, which provides the first step in determining an upper bound on s for all the vertex-primitive s-arc-transitive digraphs.

Introduction

A digraph Γ is a pair $(V,\to )$ with a set V (of vertices) and an antisymmetric irreflexive binary relation → on V. For a non-negative integer s, an s-arc of Γ is a sequence $v_0,v_1,\dots ,v_s$ of vertices with $v_i\to v_{i+1}$ for each $i=0,\dots ,s-1$. A 1-arc is also simply called an arc. For a subgroup G of $\mathrm{Aut}(\mathrm{\Gamma })$, we say Γ is $(G,s)$-arc-transitive if G acts transitively on the set of s-arcs of Γ. An $(\mathrm{Aut}(\mathrm{\Gamma }),s)$-arc-transitive digraph Γ is said to be s-arc-transitive. Note that a vertex-transitive $(s+1)$-arc-transitive digraph is necessarily s-arc-transitive. A transitive permutation group G on a set Ω is said to be primitive if G does not preserve any nontrivial partition of Ω. For a subgroup G of $\mathrm{Aut}(\mathrm{\Gamma })$, we say Γ is G-vertex-primitive if G is primitive on the vertex set. An $\mathrm{Aut}(\mathrm{\Gamma })$-vertex-primitive digraph Γ is said to be vertex-primitive. All digraphs and groups considered in this paper will be finite.

It appears that vertex-primitive s-arc-transitive digraphs with large s are very rare. Indeed, the existence of vertex-primitive 2-arc-transitive digraphs besides directed cycles was only recently determined [8] and no vertex-primitive 3-arc-transitive examples are known. In [9] the authors asked the following question:

Question 1.1

Is there an upper bound on s for vertex-primitive s-arc-transitive digraphs that are not directed cycles?

A group G is said to be almost simple if G has a unique minimal normal subgroup T and T is a nonabelian simple group. These are precisely the groups lying between a nonabelian simple group T and its automorphism group $\mathrm{Aut}(T)$. A systematic investigation of the O'Nan-Scott types of primitive groups has reduced Question 1.1 to almost simple groups by showing that an upper bound on s for vertex-primitive s-arc-transitive digraphs Γ with $\mathrm{Aut}(\mathrm{\Gamma })$ almost simple will be an upper bound on s for all vertex-primitive s-arc-transitive digraphs [9, Corollary 1.6]. This paper provides the first step in determining such an upper bound by studying vertex-primitive s-arc-transitive digraphs whose automorphism group is an almost simple linear group. Our main result is as follows.

Theorem 1.2

Let Γ be a G-vertex-primitive $(G,s)$-arc-transitive digraph, where G is almost simple with socle ${\mathrm{PSL}}_n(q)$. Then $s⩽2$.

We remark that an infinite family of G-vertex-primitive $(G,2)$-arc-transitive digraphs with $G={\mathrm{PSL}}_3(p^2)$ for each prime $p>3$ such that $p\equiv \pm 2(\mathrm{mod}5)$ was constructed in [8]. These digraphs have vertex stabilizer ${\mathrm{A}}_6$ and arc-stabilizer ${\mathrm{A}}_5$, and are the only known examples of G-vertex-primitive $(G,2)$-arc-transitive digraphs such that G is almost simple. A complete classification of G-vertex-primitive $(G,2)$-arc-transitive digraphs for almost simple groups G, even for those with $\mathrm{Soc}(G)={\mathrm{PSL}}_n(q)$, seems out of reach at this stage, though would be achievable for small values of n.

Note that if $\mathrm{Soc}(G)={\mathrm{PSL}}_n(q)$ then either $G⩽{\mathrm{P}\mathrm{\Gamma }\mathrm{L}}_n(q)$ or G has an index 2 subgroup contained in ${\mathrm{P}\mathrm{\Gamma }\mathrm{L}}_n(q)$ and G contains an element that acts on the projective space associated with G by interchanging the set of 1-spaces and the set of hyperplanes. For any G-vertex-primitive $(G,s)$-arc-transitive digraph Γ, the vertex stabilizer $G_v$ for any vertex v of Γ is maximal in G. We prove Theorem 1.2 by analyzing the maximal subgroups of G according to the classes provided by Aschbacher's theorem [1]. The classes ${\mathcal{C}}_1$, ${\mathcal{C}}_2$, …, ${\mathcal{C}}_8$ are discussed in Sections 4–6, while the remaining class ${\mathcal{C}}_9$ is dealt with in Section 3. We actually prove that there is no G-vertex-primitive $(G,2)$-arc-transitive digraph with $G_v$ from classes ${\mathcal{C}}_3$, …, ${\mathcal{C}}_6$ (Theorem 5.6) though the possibility for an example with $G_v$ from classes ${\mathcal{C}}_1$, ${\mathcal{C}}_2$, ${\mathcal{C}}_7$ or ${\mathcal{C}}_8$ remains open. The examples in [8] have $G_v$ from the class ${\mathcal{C}}_9$. At the end of Section 6 we give a proof of Theorem 1.2.