Definably simple stable groups with finitary groups of automorphisms

TL;DR

This paper classifies certain infinite simple groups with stability properties, showing they are Lie type groups over finite fields, and explores automorphisms that resemble Frobenius maps to classify specific stable groups.

Contribution

It provides a classification of infinite definably simple stable groups with automorphisms similar to Frobenius maps, linking them to Lie type groups over finite fields.

Findings

Infinite definably simple locally finite groups are Lie type groups over finite fields.

Automorphisms resembling Frobenius maps help classify certain stable groups.

Conditions on automorphisms determine the structure of definably simple stable groups.

Abstract

We prove that infinite definably simple locally finite groups of finite centraliser dimension are simple groups of Lie type over locally finite fields. Then, we identify conditions on automorphisms of a stable group that make it resemble the Frobenius maps, and allow us to classify definably simple stable groups in the specific case when they admit such automorphisms.