Classification theory of simple locally finite groups

Abstract

This thesis constitutes a contribution to applied stability theory. We consider the classification problem of the stable simple locally finite groups.

First the classification of the finite simple groups is used to reduce the problem to an identification problem for the simple locally finite groups of Lie type and an interpretation problem in model theoretic algebra.

In chapter three, the identification problem is solved. It is shown that the union of a chain of groups of the same Lie type over finite fields is a group of Lie type over a locally finite field. This result, together with the classification of the finite simple groups, implies that an infinite simple periodic linear group is a group of Lie type over a locally finite field.

The next two chapters solve the interpretation problem, and complete the proof that a stable simple locally finite group is a Chevalley group over an algebraically closed field. We also show that the class of Chevalley groups of a fixed Lie type is finitely axiomatisable.

Chapter six contains a partial classification of the nonsoluble locally finite groups of finite Morley rank.

In the final chapter, we show that a simple constructible group over an algebraically closed field is a Chevalley group. The proof is model theoretic, and makes no use of algebraic geometry or Lie algebras. This result can be regarded as a nonstandard corollary of the classification of the finite simple groups.