Binary permutation groups: alternating and classical groups

TL;DR

This paper introduces a novel approach using 'beautiful subsets' to analyze finite binary permutation groups, successfully proving Cherlin's conjecture for groups with alternating socles and certain classical groups.

Contribution

It presents a new method involving 'beautiful subsets' to determine binary actions, resolving many open cases of Cherlin's binary groups conjecture.

Findings

Proves Cherlin's binary groups conjecture for groups with alternating socles.

Establishes the effectiveness of 'beautiful subsets' in analyzing binary actions.

Provides a framework applicable to classical groups in permutation group theory.

Abstract

We introduce a new approach to the study of finite binary permutation groups and, as an application of our method, we prove Cherlin's binary groups conjecture for groups with socle a finite alternating group, and for the $\mathcal{C}_1$-primitive actions of the finite classical groups. Our new approach involves the notion, defined with respect to a group action, of a `\emph{beautiful subset}'. We demonstrate how the presence of such subsets can be used to show that a given action is not binary. In particular, the study of such sets will lead to a resolution of many of the remaining open cases of Cherlin's binary groups conjecture.