Recognizing $\operatorname{PGL}_3$ via generic $4$-transitivity

TL;DR

This paper proves that the only highly symmetric action of a finite Morley rank group on a rank 2 set is the natural projective linear group action on the plane, characterizing $ ext{PGL}_3$ uniquely.

Contribution

It establishes a uniqueness result for the natural action of $ ext{PGL}_3$ among transitive, generically 4-transitive actions on rank 2 sets in the context of finite Morley rank groups.

Findings

$ ext{PGL}_3$ is uniquely characterized by its 4-transitivity on the projective plane.

The paper rules out other groups with similar transitivity properties in this setting.

The result deepens understanding of symmetry actions in groups of finite Morley rank.

Abstract

We show that the only transitive and generically $4$-transitive action of a group of finite Morley rank on a set of Morley rank $2$ is the natural action of $\operatorname{PGL}_3$ on the projective plane.