Graphs with subconstituents containing $L_{3}(p)$

Abstract:

Let $\Gamma$ be a finite connected undirected graph, $G$ a vertex-transitive subgroup of $\operatorname {aut} (\Gamma )$, $\{ x,y\}$ an edge of $\Gamma$ and ${G_i}(x,y)$ the subgroup of $G$ fixing every vertex at a distance of at most $i$ from $x$ or $y$. We show that if the stabilizer ${G_x}$ contains a normal subgroup inducing ${L_3}(p)$, $p$ a prime, on the set of vertices adjacent to $x$, then ${G_5}(x,y) = 1$.