Two-closure of rank 3 groups in polynomial time

TL;DR

This paper presents a polynomial-time algorithm for computing the 2-closure of rank 3 permutation groups, which are characterized by having exactly three orbits on pairs of elements.

Contribution

The paper introduces the first polynomial-time algorithm to compute the 2-closure of rank 3 groups from their generators, advancing computational group theory.

Findings

Algorithm runs in polynomial time

Successfully computes 2-closure from generators

Applicable to rank 3 permutation groups

Abstract

A finite 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 $G$. We construct a polynomial-time algorithm which given generators of a rank 3 group computes generators of its 2-closure.