Highly Arc-Transitive Digraphs

Vertex primitive case

Question

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

By O’Nan-Scott-Praeger theorem, there are eight types of finite primitive groups. For four of the eight types, a quasiprimitive group of that type has a normal regular subgroup. Praeger showed that if $\Gamma$ is a $(G,2)$-arc-transitive digraph and $G$ has a normal subgroup that acts regularly on $V$, then $\Gamma$ is a directed cycle. Thus it remains consider the four remaining types: AS, SD, CD and PA. Giudici and Xia showed that groups of type SD, CD and PA do not have nontrivial $3$-arc-transitive digraphs. So the question is reduced to almost simple primitive groups.

Vertex imprimitive case

However, for imprimitive case, Praeger shows that there are infinitely many examples of a finite $k$-regular digraph with a group of automorphisms acting transitively on $s$-arcs (but not on $(s+1)$-arcs).

Let $\Gamma$ be a digraph and $G=\mathrm{Aut}(\Gamma )$. Let $A$ be the set of arcs. If $(u,v)\in A$, then $u$ is an in-neighbor of $v$, and $v$ is an out-neighbor of $u$. The set of in-neighbors of a vertex $v$ of $\Gamma$ is denoted ${\Gamma }^{-}(v)$ and the set of out-neighbors is denoted ${\Gamma }^{+}(v)$.

If $\Gamma$ is $G$-vertex-transitive, then the in-local action of $(\Gamma ,G)$ is the permutation group induced by the vertex-stabilizer $G_v$ on ${\Gamma }^{-}(v)$. The out-local action is defined analogously.

Definition

We say that two permutation groups $L_{+}$ and $L_{-}$ are compatible if there exists a finite $G$-vertex-transitive digraph $\Gamma$ such that $(\Gamma ,G)$ has in-local action $L_{-}$ and out-local action $L_{+}$.

In fact, ^shmliq and ^rkwv8u are equivalent, see section 3.

subconstituents

Let $G$ be a transitive permutation group on a set $V$, let $u,v\in V$ and let $A$ be the orbit of $(u,v)$ under $G$. The set $A$ is an orbital of $G$ and the digraph $\Gamma$ with vertex set $V$ and $\mathrm{arc}$ set $A$ is called an orbital digraph.

Clearly, $\Gamma$ is both $G$-vertex-transitive and $G$-arctransitive. The in- and out-local actions of $(\Gamma ,G)$ are called paired subconstituents of $G$ (with respect to $A$).

Problem: determine which pairs of permutation groups can arise as paired subconstituents of a finite permutation group.

Ding’s work

Definition

We say that two groups $L_1$ and $L_2$ are compatible if there exists a group $G$ with isomorphic normal subgroups $N_1$ and $N_2$ such that $L_1\cong G/N_1$ and $L_2\cong G/N_2$.

Theorem

Compatible groups with no abelian composition factors have compatible normal series.