# Residually nilpotent group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a variation of nilpotence|Find other variations of nilpotence | Read a survey article on varying nilpotence

## Contents

## Definition

A group is termed **residually nilpotent** if it satisfies the following equivalent conditions:

- Given any non-identity element, there is a normal subgroup not containing that element, such that the quotient group is nilpotent
- The lower central series reaches the identity element at or before the stage; in other words, the intersection of all the terms of the (finite) lower central series is the trivial group.
- The nilpotent residual of the group is the trivial subgroup.

## Formalisms

### In terms of the residually operator

This property is obtained by applying the residually operator to the property: nilpotent group

View other properties obtained by applying the residually operator

## Relation with other properties

### Stronger properties

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

nilpotent group | lower central series reaches identity in finite many steps | |FULL LIST, MORE INFO | ||

free group | free on some generating set | free implies residually nilpotent | (any nontrivial finite nilpotent group is residually nilpotent but clearly not free) | |FULL LIST, MORE INFO |

residually finite p-group |

### Weaker properties

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

Hypocentral group | |FULL LIST, MORE INFO | |||

Residually solvable group | |FULL LIST, MORE INFO | |||

Hypoabelian group | Residually solvable group|FULL LIST, MORE INFO |

### Incomparable properties

- Hypercentral group: A residually nilpotent group need not have its upper central series go towards the group. In fact, free groups are examples of centerless residually nilpotent groups.

## Metaproperties

### Direct products

This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property

View other finite direct product-closed group properties

A finite direct product of residually nilpotent groups is residually nilpotent.