Quasiprimitive and bi-quasiprimitive highly-arc-transitive digraphs and finite simple groups

We extend the notion of an $H$-normal quotient digraph of an $H$-vertex-transitive digraph to that of an $H$-subnormal quotient digraph. Using these concepts, together with bipartite halves of bipartite digraphs, we show that, for each finite connected $H$-vertex-transitive, $(H,s)$-arc-transitive digraph with $s\geqslant6$, either some $H$-normal quotient is a directed cycle of length at least $3$, or there is an $(L,t)$-arc-transitive digraph with $t\geqslant (s-3)/2$, and $L$ a vertex-quasiprimitive almost simple group with socle a composition factor of $H$. This connection demonstrates that, to understand finite $s$-arc-transitive digraphs with large $s$, those admitting a vertex-quasiprimitive almost simple $s$-arc-transitive subgroup of automorphisms play a central role. We show that for each $s$ and each odd valency $k$, there are infinitely many $(H,s)$-arc-transitive digraphs of valency $k$ with $H$ a finite alternating group. In addition we discovered a novel construction which takes as input a connected non-bipartite $H$-vertex-transitive, $(H,s)$-arc-transitive digraph, and outputs a connected bipartite $G$-vertex-transitive, $(G,2s)$-arc-transitive digraph with $G=(H\times H).2$. This leads to construction of vertex-bi-quasiprimitive $s$-arc-transitive digraphs, for arbitrarily large $s$. Our investigations yield several new open problems.

💡 Research Summary

The paper investigates finite highly‑arc‑transitive digraphs, focusing on the case where the arc‑transitivity degree $s$ is at least 6. The authors first generalise the classical notion of an $H$‑normal quotient digraph to an $H$‑subnormal quotient digraph. This broader concept allows one to consider quotients by subnormal subgroups rather than only normal ones, and it is crucial for the reduction arguments that follow.

The central result (Theorem 1.2) states that for any finite connected $H$‑vertex‑transitive $(H,s)$‑arc‑transitive digraph $\Gamma$ with $s\ge6$, exactly one of two alternatives must occur: (i) some $H$‑normal quotient of $\Gamma$ is a directed cycle $C_r$ with $r\ge3$, or (ii) there exists a group $L$ that is almost simple, acts vertex‑quasiprimitively on the vertex set, and for which a $(L,t)$‑arc‑transitive digraph exists with $t\ge (s-3)/2$. In other words, any highly‑arc‑transitive digraph of large $s$ can be reduced, via a finite sequence of normal, subnormal, or bipartite‑half operations, to a digraph whose automorphism group is almost simple and quasiprimitive. The simple socle of $L$ is a composition factor of the original group $H$, showing that the structure of $H$ essentially collapses to a single non‑abelian simple group in the “core” of the construction.

Using this reduction framework, the authors produce several infinite families of examples. First, they show that for every odd valency $k$ and any $s\ge1$, there are infinitely many $(H,s)$‑arc‑transitive digraphs of valency $k$ with $H$ a finite alternating group $A_n$ (or symmetric group $S_n$). These digraphs have no non‑trivial cyclic normal quotients, demonstrating that the second alternative of Theorem 1.2 indeed occurs.

The most novel construction is a “doubling’’ process (Construction 5.1). Starting from any connected non‑bipartite $H$‑vertex‑transitive $(H,s)$‑arc‑transitive digraph $\Delta$, the authors build a new bipartite digraph $\Gamma$ on which $G=(H\times H).2$ acts. The resulting digraph is $(G,2s)$‑arc‑transitive and vertex‑bi‑quasiprimitive (i.e., each non‑trivial normal subgroup of $G$ has exactly two equal‑size orbits). This construction yields vertex‑bi‑quasiprimitive $s$‑arc‑transitive digraphs for arbitrarily large $s$, and it can be iterated to obtain families of arbitrarily high arc‑transitivity while preserving the bi‑quasiprimitive property.

The paper also discusses the classification of quasiprimitive groups that can appear in this context. It reviews the known types (AS, PA, SD, CD) and shows that for $s\ge3$ only the almost‑simple (AS) and product‑action (PA) types can give rise to $s$‑arc‑transitive digraphs after the reduction process. In particular, the authors argue that 3‑arc‑transitive examples with a vertex‑primitive almost‑simple group are unlikely, supporting existing conjectures that such examples do not exist beyond $s=2$.

Finally, the authors pose several open problems: (1) determining for which non‑abelian simple groups $T$ there exist vertex‑quasiprimitive $(G,s)$‑arc‑transitive digraphs with $T$ as the socle of $G$; (2) whether $P!A$‑type quasiprimitive actions can arise on digraphs whose vertex set is not a direct product $V_0^m$; (3) the existence of bi‑quasiprimitive $s$‑arc‑transitive digraphs for $s=4,5$ that evade the reduction to an almost‑simple group; and (4) a full description of “basic” digraphs whose only normal quotients are directed cycles. These questions point to a rich interplay between permutation group theory and the combinatorial structure of highly‑arc‑transitive digraphs, suggesting many directions for future research.