Exceptional groups and the $s$-arc-transitivity of vertex-primitive digraphs, I

In this paper, we study the primitive actions of almost simple exceptional groups of Lie type on (s)-arc-transitive digraphs. Our motivation is the following question posed by Giudici and Xia: Is there an upper bound on $s$ for finite vertex-primitive $s$-arc-transitive digraphs that are not directed cycles? In a 2018 paper, Giudici and Xia reduced this question to the case where the automorphism group of the digraph is an almost simple group with socle (L). Subsequently, it has been proved that $s\leq 2$ when (L) is a linear, symplectic or alternating group, and $s\leq 1$ when (L) is a Suzuki group, a small Ree group, or one of $22$ specific sporadic groups. In this paper, we prove that $s\leq 2$ when (L) is $ {}^3D_4(q)$, $G_2(q)$ (including $G_2(2)’$), ${}^2F_4(q)$ (including ${}^2F_4(2)’$), $F_4(q)$, $E_6(q)$ or ${}^2E_6(q)$.

💡 Research Summary

The paper addresses a long‑standing question posed by Giudici and Xia (2018): whether there exists an absolute upper bound on the parameter s for finite vertex‑primitive s‑arc‑transitive digraphs that are not directed cycles. Giudici and Xia reduced the problem to the case where the automorphism group of the digraph is almost simple with socle L. Earlier work settled the bound for linear, symplectic, alternating, Suzuki, small Ree, and many sporadic groups, showing s ≤ 2 in the first three families and s ≤ 1 in the latter cases. The present work completes the analysis for the remaining families of exceptional groups of Lie type.

The authors consider an almost simple group H with socle L belonging to the set { ³D₄(q), G₂(q), ²F₄(q), F₄(q), E₆(q), ²E₆(q) } (including the derived subgroups G₂(2)′ and ²F₄(2)′). They prove that any connected, vertex‑primitive, s‑arc‑transitive digraph Γ on which H acts as a group of automorphisms must satisfy s ≤ 2. No concrete example with s = 2 is known, and the authors remark that constructing such a digraph remains an open problem.

The proof proceeds through a blend of group‑theoretic analysis and computational verification. The key steps are:

1. Factorisation of vertex stabilisers – Using a result of Giudici and Xia, the authors show that for a 2‑arc (u→v→w) in Γ, the vertex stabiliser H_v must factor as H_{uv} H_{vw}. This factorisation must be homogeneous and core‑free, imposing strong restrictions on the structure of H_v.

2. Structure of H_v – By examining the radical and the quasisimple components of H_v, Lemma 2.6 classifies the possible simple composition factors that can appear. The possibilities reduce to a short list: Sp₄(2^f), PΩ⁺₈(q), PSL₂(q), and several small almost simple groups listed in Table 1.

3. Use of maximal subgroup classifications – The authors exploit the known classifications of maximal subgroups for the exceptional groups under consideration (citing works of Kleidman, Liebeck, Seitz, and recent extensions by Craven). This allows them to enumerate all candidate subgroups that could serve as H_{uv} or H_{vw}.

4. Prime‑divisor arguments – Applying Zsigmondy’s theorem, they guarantee the existence of primitive prime divisors of qⁿ − 1 for suitable n. These primes appear in the orders of the relevant subgroups and force contradictions when s ≥ 3, because the required homogeneous factorisations cannot accommodate the large prime powers.

5. Computational verification – A Magma routine (facsm) is employed to generate all factorisations H = AB with prescribed divisibility conditions. The routine confirms that no admissible factorisation exists that would allow s ≥ 3 for the groups ³D₄(q), G₂(q), and ²F₄(q). For the larger groups F₄(q), E₆(q), and ²E₆(q), the theoretical restrictions already rule out s ≥ 3.

6. Bounding s via normal subgroup actions – Lemmas 2.13 and 2.14 show that if the vertex stabiliser contains a large abelian normal subgroup C₂^m·O, then the existence of a primitive prime divisor r with certain order conditions forces s ≤ 2 (or s ≤ 1 when the stabiliser is cyclic). This argument eliminates the possibility of higher‑arc transitivity in all remaining cases.

The main theorem (Theorem 1.1) thus follows: for any connected, vertex‑primitive, s‑arc‑transitive digraph Γ whose automorphism group is almost simple with socle among the listed exceptional groups, we have s ≤ 2. The paper notes that the cases of E₇(q) and E₈(q) remain open because a complete classification of maximal subgroups for those groups is not yet available.

In summary, the work completes the classification of possible s‑arc‑transitivity levels for vertex‑primitive digraphs under almost simple groups, extending the bound s ≤ 2 to all exceptional groups of Lie type except the two largest families. It combines deep results from the theory of finite simple groups, detailed analysis of subgroup structures, and computer‑assisted verification, thereby providing a comprehensive answer to the Giudici‑Xia question for the majority of the remaining families.