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.

DOI: https://doi.org/10.1017/S0305004118000403

Abstract

A permutation group $G$ on a set $\Omega$ is said to be binary if, for every $n\in\mathbb{N}$ and for every $I,J \in \Omega^n$, the $n$-tuples $I$ and $J$ are in the same $G$-orbit if and only if every pair of entries from $I$ is in the same $G$-orbit to the corresponding pair from $J$. 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 $\Sym(n)$ with their natural action, the groups of prime order, and the affine groups $V\rtimes O(V)$ where $V$ 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 $\SLq$, $\suzuki$, $\ree$ or $\PSU_3(q)$. Our method uses the notion of a ``strongly non-binary action''.

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