An important class of examples is provided by groups of finite Morley rank (see below).
Progress towards this conjecture has followed Borovik’s program of transferring methods used in classification of finite simple groups.
One possible source of counterexamples is bad groups: nonsoluble connected groups of finite Morley rank all of whose proper connected definable subgroups are nilpotent.
(A group is called connected if it has no definable subgroups of finite index other than itself.)
A number of special cases of this conjecture have been proved; for example: