Rank three innately transitive permutation groups and related $2$-transitive groups

TL;DR

This paper classifies finite innately transitive permutation groups, extending known classifications of primitive and quasiprimitive groups, and introduces new infinite families and sporadic examples, advancing the understanding of their structure.

Contribution

It provides a comprehensive classification of finite innately transitive groups, including new examples and the analysis of special pairs in 2-transitive groups.

Findings

Identified three infinite families of innately transitive groups.

Discovered three sporadic examples of such groups.

Determined configurations called special pairs in 2-transitive groups.

Abstract

The sets of primitive, quasiprimitive, and innately transitive permutation groups may each be regarded as the building blocks of finite transitive permutation groups, and are analogues of composition factors for abstract finite groups. This paper extends classifications of finite primitive and quasiprimitive groups of rank at most $3$ to a classification for the finite innately transitive groups. The new examples comprise three infinite families and three sporadic examples. A necessary step in this classification was the determination of certain configurations in finite almost simple $2$-transitive groups called special pairs.