A New Trichotomy Theorem

TL;DR

This paper proves a new structural property of minimal counterexamples to a major conjecture in group theory, narrowing the possible types of simple groups of finite Morley rank.

Contribution

It establishes a bound on the normal 2-rank of minimal counterexamples, extending the trichotomy theorem without tameness assumptions.

Findings

Minimal counterexamples have normal 2-rank at most two

No groups exist strictly between the generic and quasithin cases

Supports the classification of simple groups of finite Morley rank

Abstract

We show that a minimal counter example to the Cherlin-Zilber Algebraicity Conjecture for simple groups of finite Morley rank has normal 2-rank at most two, which is a tameness free version of Borovik's original trichotomy theorem. This result serves as a bridge by showing that there are no groups found strictly between the generic and quasithin cases, i.e. between groups of Lie rank at least three, and groups of Lie rank one and two. Again this result depends upon previous work for the uniqueness case analysis.