# Nilpotent normal subgroup

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property (itself viewed as a subgroup property): nilpotent group

View a complete list of such conjunctions

## Contents

## Definition

### Symbol-free definition

A subgroup of a group is termed a **nilpotent normal subgroup** if it is nilpotent as a group, and normal as a subgroup.

## Relation with other properties

### Stronger properties

- Cyclic normal subgroup
- Central subgroup (subgroup contained in the center)
- Abelian normal subgroup
- Subgroup generated by Abelian normal subgroups

### Weaker properties

## Facts

The subgroup of a group generated by all its nilpotent normal subgroups is termed the Fitting subgroup, and if a group equals its Fitting subgroup, then it is termed a Fitting group. For finite groups, the Fitting subgroup is the largest nilpotent normal subgroup, and a Fitting group is the same thing as a nilpotent group.