**TASK V - SIMPLE GROUPS OF FINITE MORLEY RANK**

**
Description**

Morley rank is a
model-theoretic notion of
dimension. In the case of
algebraic groups over algebraically closed fields, it coincides with algebraic
dimension. The Cherlin-Zilber conjecture states that a simple (infinite) group
of finite Morley rank
is an algebraic group over an algebraically closed field, and is a
central goal.

Methods of the classification of finite simple groups
can
be applied successfully to those infinite groups possessing a
reasonable concept of
dimension (such as Morley rank), and one attempts
to relate the `generic finite simple groups' to simple groups of finite
Morley rank.

**
List of specific problems**

**V.1. Simple groups of finite Morley rank**

a) Complete the classification in even characteristic case.

b) In odd type case, reduce the analysis
to certain specific configurations under inductive assumptions; eliminate
the inductive assumptions. Eliminate simple groups with large finite Sylow
2-subgroups.

c) Prove that bad fields in positive
characteristic do not exist, if necessary under a plausible number-theoretic
hypothesis, via a development of Lang-Weil estimates for definable sets in
a locally finite bad field.

d) Apply FMR techniques in finite group theory.** **