Sharp permutation groups whose point stabilizers are Frobenius groups with cyclic Frobenius kernel

Abstract

Let ( G , X ) be a transitive non-geometric sharp permutation group of type { 0 , k } and let x ∈ X . We prove that if the point stabilizer G x is a Frobenius group with cyclic Frobenius kernel, then G x ≅ AGL ( 1 , p ) for some odd prime p and ( G , X ) is permutation isomorphic to a certain subgroup of AGL ( 2 , p ) acting on its natural module.