Binary permutation groups: Alternating and classical groups

Gill, Nick and Spiga, Pablo (2020). Binary permutation groups: Alternating and classical groups. American Journal of Mathematics, 142(1) pp. 1–43.

DOI: https://doi.org/10.1353/ajm.2020.0000

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.

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