Nilpotent-quotient subgroup

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Definition

A subgroup of a group is termed a nilpotent-quotient normal subgroup if it is a normal subgroup and the quotient group is a nilpotent group.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Abelian-quotient normal subgroup
Upward-closed normal subgroup

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Normal subgroup

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.