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?)