Permutation groups with projective unitary subconstituents

Abstract

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