# Nilpotent-quotient subgroup

From Groupprops

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]

## Contents

## 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 | |

normal subgroup | |FULL LIST, MORE INFO |

## Facts

The intersection of all nilpotent-quotient normal subgroups is termed the nilpotent residual, and this is also described as the 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.