Involutive automorphisms of N∘∘N_\circ^\circ groups of finite Morley rank

We classify a large class of small groups of finite Morley rank: $N_\circ^\circ$-groups which are the infinite analogues of Thompson's $N$-groups. More precisely, we constrain the $2$-structure of groups of finite Morley rank containing a definable, normal, non-soluble, $N_\circ^\circ$-subgroup

Simple groups of Morley rank 5 are bad

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 $5$

Rank 3 Bingo

We classify irreducible actions of connected groups of finite Morley rank on abelian groups of Morley rank 3