Pseudofinite groups as fixed points in simple groups of finite Morley rank

TL;DR

This paper characterizes fixed point groups of automorphisms in simple groups of finite Morley rank, showing they are extensions of Chevalley groups over pseudofinite fields, and classifies certain pseudofinite groups.

Contribution

It establishes a link between fixed points of automorphisms in finite Morley rank groups and pseudofinite Chevalley groups, providing a classification of specific pseudofinite groups.

Findings

Fixed points are extensions of Chevalley groups over pseudofinite fields

Classification of non-abelian definably simple pseudofinite groups of finite centralizer dimension

Connection between automorphism fixed points and pseudofinite group structures

Abstract

We prove that if the group of fixed points of a generic automorphism of a simple group of finite Morley rank is pseudofinite, then this group is an extension of a (twisted) Chevalley group over a pseudofinite field. On the way to obtain this result, we classify non-abelian definably simple pseudofinite groups of finite centralizer dimension (using the ideas of John S. Wilson).