Nilpotent normal subgroup

Definition
A subgroup of a group is termed a nilpotent normal subgroup if it is nilpotent as a group, and normal as a subgroup.

Facts
The subgroup of a group generated by all its nilpotent normal subgroups is termed the Fitting subgroup, and if a group equals its Fitting subgroup, then it is termed a Fitting group. For finite groups, the Fitting subgroup is the largest nilpotent normal subgroup, and a Fitting group is the same thing as a nilpotent group.