# Normal subgroup of nilpotent group

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property imposed on theambient group: nilpotent group

View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

## Contents

## Definition

The term **normal subgroup of nilpotent group** is used for a subgroup of a group where the whole group is a nilpotent group and the subgroup is a normal subgroup.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

subgroup of abelian group | ||||

normal subgroup of group of prime power order | ||||

characteristic subgroup of nilpotent group |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

normal subgroup | |FULL LIST, MORE INFO | |||

normal subgroup contained in hypercenter | |FULL LIST, MORE INFO | |||

nilpotent-quotient subgroup | normal subgroup such that the quotient group is a nilpotent group. | |FULL LIST, MORE INFO | ||

normal subgroup satisfying the subgroup-to-quotient powering-invariance implication | if the whole group and the normal subgroup are powered over a prime, so is the quotient group. | two proofs: via normal subgroup contained in the hypercenter via nilpotent-quotient |
Normal subgroup contained in the hypercenter|FULL LIST, MORE INFO | |

subgroup of nilpotent group | |FULL LIST, MORE INFO |