Linear groups of finite Morley rank

TL;DR

This paper proves that certain non-algebraic simple groups of finite Morley rank either have no involutions or resemble bad groups, supporting the Cherlin-Zilber conjecture in this context.

Contribution

It establishes the absence of involutions in non-algebraic simple groups of finite Morley rank with definable field representations, advancing the understanding of their structure.

Findings

Non-algebraic simple groups with definable representations have no involutions.

Such groups either resemble bad groups or are algebraic.

Supports the Cherlin-Zilber conjecture for these groups.

Abstract

We show that a non-algebraic simple group of finite Morley rank with a definable representation over a field has no involutions, and otherwise resembles a bad group. In particular, the modern form of the Cherlin-Zilber alebaricity conjecture hold for such groups.