On 2-closures of rank 3 groups

TL;DR

This paper characterizes the 2-closures of rank 3 permutation groups, showing that for certain affine groups, the 2-closure retains the affine and one-dimensional structure, advancing understanding of permutation group closures.

Contribution

It provides a detailed description of 2-closures of rank 3 groups and proves that large primitive affine rank 3 groups have affine, one-dimensional 2-closures.

Findings

Describes 2-closures of rank 3 groups.

Proves 2-closure of large primitive affine rank 3 groups is affine and one-dimensional.

Enhances understanding of the structure of permutation group closures.

Abstract

A permutation group $G$ on $\Omega$ is called a rank 3 group if it has precisely three orbits in its induced action on $\Omega \times \Omega$. The largest permutation group on $\Omega$ having the same orbits as $G$ on $\Omega \times \Omega$ 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 2-closure of a primitive one-dimensional affine rank 3 permutation group of sufficiently large degree is also affine and one-dimensional.