Cherlin's Conjecture that an infinite simple group of finite Morely rank is an algebraic group over an algebraically closed field has been around for many years now, and has been the starting point for a considerable amount of research. In this survey paper we describe two approaches towards Cherlin's Conjecture, first without any stability assumption via the theory of algebraic groups and secondly via the theory of Tits buildings in the context of finite Morley rank. While the conjecture is still open, our results cover most classes of classical and algebraic groups and the (twisted) Chevalley groups.

Algebraic groups and Cherlin's Conjecture

Restricted to algebraic groups, Cherlin's Conjecture reduces to the following: If the group G(k) of k-rational points of an algebraic group G is a simple group of finite Morley rank, does this imply that k is algebraically closed? In similar form this was asked by Borovik and Nesin in [BN], p.367.

We show that this is (almost) true: whenever G is almost simple (or k-simple) and k-isotropic, the field k (or a finite extension of k) is definable in the pure group structure of G(k); hence if such a group has finite Morley rank, then k has to be either algebraically closed or real closed, thus answering Questions B.46 and B.48 in [BN] for these cases. This result should come as no surprise since in [BT73] it is shown that the group of abstract automorphisms of such a groups is essentially the group itself extended by the automorphism group of the field.