A Jordan decomposition for groups of finite Morley rank

TL;DR

This paper establishes a Jordan decomposition for certain simple groups of finite Morley rank, providing a detailed structural understanding of their Borel subgroups and classifying these groups through a Tetrachotomy theorem.

Contribution

It introduces a Jordan decomposition theorem for minimal connected simple groups of finite Morley rank with non-trivial Weyl group and offers a new classification via Tetrachotomy.

Findings

Proved a Jordan decomposition theorem for these groups

Provided a structural description of Borel subgroups

Classified minimal connected simple groups into four types

Abstract

We prove a Jordan decomposition theorem for minimal connected simple groups of finite Morley rank with non-trivial Weyl group. From this, we deduce a precise structural description of Borel subgroups of this family of simple groups. Along the way we prove a Tetrachotomy theorem that classifies minimal connected simple groups. Some of the techniques that we develop help us obtain a simpler proof of a theorem of Burdges, Cherlin and Jaligot.