Application of maximal subgroups of exceptional groups in s-arc-transitivity of vertex-primitive digraphs (I)
Fu-Gang Yin, Lei Chen
TL;DR
This work addresses Giudici and Xia's question on whether there exists a universal bound on $s$ for vertex-primitive $s$-arc-transitive digraphs beyond directed cycles by analyzing almost-simple automorphism groups with socles among exceptional groups. The authors leverage Craven's complete maximal-subgroup lists for $F_{4}(q)$, $E_{6}(q)$, and ${}^{2}E_{6}(q)$ to reduce the problem to group-factorisation and orbit structure in each case, employing Magma-assisted computations and root-system analyses for subgroups of maximal rank and parabolic subgroups. They prove that in the families ${}^{3}D_{4}(q)$, $G_{2}(q)$, ${}^{2}F_{4}(q)$, $F_{4}(q)$, $E_{6}(q)$, and ${}^{2}E_{6}(q)$, any connected vertex-primitive $(H,s)$-arc-transitive digraph with $s\ge 2$ must satisfy $s\le 2$. This partial resolution advances the understanding of upper bounds on $s$ for a broad class of exceptional groups and sets the stage for treating remaining families (e.g., $E_{7}(q)$ and $E_{8}(q)$) in future work. The techniques combine structural group theory (factorisations, maximal rank subgroups, parabolic analysis) with computational algebra (Magma) to rule out all potential stabilisers that could yield $s\ge 3$.
Abstract
The investigation of maximal subgroups of simple groups of Lie type is intimately related to the study of primitive actions. With the recent publication of Craven's paper giving the complete list of the maximal subgroups of \(F_{4}(q)\), \(E_{6}(q)\) and \({}^{2}E_{6}(q)\), we are able to thoroughly analyse the primitive action of an exceptional group on an \(s\)-arc-transitive digraph and partially answer 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? Giudici and Xia reduced this question to the case where the automorphism group of the digraph is an almost simple group. Subsequently, it was proved that $s\leq 2$ when the simple group is a projective special linear group, projective symplectic group or an alternating group, and $s\leq 1$ when the simple group is a Suzuki group, a small Ree group, or one of the 22 sporadic groups. In this work, we proved that $s\leq 2$ when the simple group is $ {}^3D_4(q)$, $G_2(q)$, ${}^2F_4(q)$, $F_4(q)$, $E_6(q)$ or ${}^2E_6(q)$.