Nilpotent-quotient subgroup

Definition
A subgroup of a group is termed a nilpotent-quotient subgroup or nilpotent-quotient normal subgroup if it is a defining ingredient::normal subgroup and the defining ingredient::quotient group is a defining ingredient::nilpotent group.

Facts
The intersection of all nilpotent-quotient normal subgroups is termed the nilpotent residual, and this is also described as the $$\omega^{th}$$ term of the transfinite lower central series It is trivial if and only if the group is a residually nilpotent group. In a finite group, the nilpotent residual is itself a nilpotent-quotient normal subgroup.