# Nilpotent-quotient subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

A subgroup of a group is termed a nilpotent-quotient subgroup or 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 subgroup
upward-closed normal subgroup

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal subgroup satisfying the subgroup-to-quotient powering-invariance implication if the whole group and subgroup are powered over a prime, so is the quotient group. nilpotent-quotient implies subgroup-to-quotient powering-invariance implication |FULL LIST, MORE INFO
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.