Introduction

A fundamental problem in the study of groups of finite Morley rank is the classification of the infinite simple ones. It was conjectured independently by Gregory Cherlin and Boris Zil'ber that they are simple algebraic groups over algebraically closed fields. This conjecture, which is not an ordinary conjecture in the sense that the classification of the finite simple groups is not an ordinary theorem, remains open. Nevertheless, in recent years there has been considerable progress in the study of some subclasses of the infinite simple groups of finite Morley rank. This progress, which uses a large number of ideas from the classification of the finite simple groups, has culminated in the following theorem:

Theorem 1.1A simple K*-group of finite Morley rank of even type is an algebraic group over an algebraically closed field of characteristic 2.

In this survey an outline of the arguments used in the proof of this result will be given.

Background

In this survey general definitions and results about groups of finite Morley rank will be mentioned only if they are needed. The reader is referred to for a good introduction to the algebraic theory, to and for discussions with emphasis on model theoretic aspects.

Nevertheless it seems useful for the reader's convenience to recall the definition of a connected component of a subgroup of a group of finite Morley rank.