Simple groups of Morley rank 5 are bad

TL;DR

This paper proves that simple groups of Morley rank 5 are 'bad' groups with all proper definable connected subgroups being nilpotent of rank at most 2, aiding classification of such groups.

Contribution

It establishes that all simple groups of Morley rank 5 are bad and classifies nonsoluble connected groups of the same rank.

Findings

Simple groups of Morley rank 5 are bad groups.

Proper definable connected subgroups are nilpotent of rank ≤ 2.

Classification of nonsoluble connected groups of Morley rank 5.

Abstract

By exploiting the geometry of involutions in $N_\circ^\circ$-groups of finite Morley rank, we show that any simple group of Morley rank $5$ is a bad group all of whose proper definable connected subgroups are nilpotent of rank at most $2$. The main result is then used to catalog the nonsoluble connected groups of Morley rank $5$.