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 properties
VIEW 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