On Primitive Extensions of Rank 3 of Symmetric Groups

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).