On Primitive Extensions of Rank 3 of Symmetric Groups

Source abstract

1. Let Ω be a finite set of arbitrary elements and let ( G , Ω ) be a permutation group on Ω . (This is also simply denoted by G ). Two permutation groups ( G , Ω) and ( G , Γ) are called isomorphic if there exist an isomorphism σ of G onto H and a one to one mapping τ of Ω onto Γ such that (g(i)) τ =g σ (i τ ) for g ∊ G and i∊Ω . For a subset Δ of Ω , those elements of G which leave each point of Δ individually fixed form a subgroup G Δ of G which is called a stabilizer of Δ . A subset Γ of Ω is called an orbit of G Δ if Γ is a minimal set on which each element of G induces a permutation. A permutation group ( G , Ω) is called a group of rank n if G is transitive on Ω and the number of orbits of a stabilizer G a of a ∊ Ω, is n . A group of rank 2 is nothing but a doubly transitive group and there exist a few results on structure of groups of rank 3 (cf. H. Wielandt [6], D. G. Higman M).