Cherlin's conjecture on finite primitive binary permutation groups

TL;DR

This paper proves Cherlin's conjecture that finite primitive binary permutation groups belong to one of three known families, focusing on the case where the group is almost simple of Lie type.

Contribution

The paper completes the proof of Cherlin's conjecture by analyzing the case of almost simple groups of Lie type.

Findings

Cherlin's conjecture is confirmed for all finite primitive binary permutation groups.

The classification includes groups of Lie type as a key case.

The proof consolidates understanding of the structure of binary permutation groups.

Abstract

A permutation group is {\it binary} if its orbits on $k$-tuples, for any integer $k\geq 2$, can be deduced from its orbits on $2$-tuples. Cherlin conjectured that a finite primitive binary permutation group $G$ must lie in one of three known families. In this paper we complete the proof of this conjecture. To do this we study the case where the group $G$ is almost simple of Lie type.