# Virtually 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

## Contents

## Definition

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

- It has a subgroup of finite index that is nilpotent.
- It has a normal subgroup of finite index that is nilpotent.
- There is a surjective homomorphism from it to a finite group such that the kernel is a nilpotent normal subgroup.
- There is a homomorphism from it to a finite group such that the kernel is a nilpotent normal subgroup.

## Formalisms

### In terms of the virtually operator

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

View other properties obtained by applying the virtually operator