Permutation groups with projective unitary subconstituents

Abstract:

Let $\Gamma$ be a finite directed graph with vertex set $V(\Gamma )$ and edge set $E(\Gamma )$ and let G be a subgroup of ${\operatorname {aut}}(\Gamma )$ which we assume to act transitively on both $V(\Gamma )$ and $E(\Gamma )$. Suppose that for some prime power q, the stabilizer $G(x)$ of a vertex x induces on both $\{ y|(x,y) \in E(\Gamma )\}$ and $\{ w|(w,x) \in E(\Gamma )\}$ a group lying between $PSU(3,{q^2})$ and $P\Gamma U(3,{q^2})$. It is shown that if G acts primitively on $V(\Gamma )$, then for each edge (x, y), the subgroup of $G(x)$ fixing every vertex in $\{ w|(x,w)$ or $(y,w) \in E(\Gamma )\}$ is trivial.