On 2-closures of rank 3 groups

Abstract

A permutation group G on Ω is called a rank 3 group if it has precisely three orbits in its induced action on Ω × Ω. The largest permutation group on Ω having the same orbits as G on Ω × Ω is called the 2-closure of G. A description of 2-closures of rank 3 groups is given. As a special case, it is proved that the 2-closure of a primitive one-dimensional affine rank 3 group of sufficiently large degree is also affine and one-dimensional.