# Cherlin's conjecture for almost simple groups of Lie rank 1

## TL;DR

This paper proves Cherlin's conjecture for certain almost simple groups of Lie rank 1, specifically those with socles isomorphic to PSL2(q), Suzuki, Ree, or PSU3(q), using the concept of strongly non-binary actions.

## Contribution

It establishes Cherlin's conjecture for a new class of almost simple groups of Lie rank 1, expanding the understanding of binary primitive permutation groups.

## Key findings

- Cherlin's conjecture holds for groups with socle PSL2(q), Suzuki, Ree, or PSU3(q).
- Introduces the notion of strongly non-binary actions to analyze group actions.
- Provides a new method to verify binary properties in permutation groups.

## Abstract

We prove Cherlin's conjecture, concerning binary primitive permutation groups, for those groups with socle isomorphic to $\mathrm{PSL}_2(q)$, ${^2\mathrm{B}_2}(q)$, ${^2\mathrm{G}_2}(q)$ or $\mathrm{PSU}_3(q)$. Our method uses the notion of a "strongly non-binary action".