Bad group

A bad group is a group $$G$$ of finite Morley rank which is non-solvable, but has the property that every proper subgroup is nilpotent. Bad groups are conjectured to not exist. This would follow from the Cherlin-Zilber conjecture.

The original plan for proving the Cherlin-Zilber conjecture was to prove that bad groups and bad fields do not exist. However, bad fields were subsequently shown to exist. (A bad field is…a field $$K$$ of finite Morley rank such that $$K^\times$$ has a definable proper subgroup?)