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

Gill, Nick; Hunt, Francis and Spiga, Pablo (2019). Cherlin's conjecture for almost simple groups of Lie rank 1. Mathematical Proceedings of the Cambridge Philosophical Society, 167(3) pp. 417–435.

## Abstract

A permutation group on a set is said to be binary if, for every and for every , the -tuples and are in the same -orbit if and only if every pair of entries from is in the same -orbit to the corresponding pair from . This notion arises from the investigation of the relational complexity of finite homogeneous structures.

Cherlin has conjectured that the only finite primitive binary permutation groups are the symmetric groups with their natural action, the groups of prime order, and the affine groups where is a vector space endowed with an anisotropic quadratic form.

We prove Cherlin's conjecture, concerning binary primitive permutation groups, for those groups with socle isomorphic to , , or . Our method uses the notion of a strongly non-binary action''.

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