Virtually nilpotent group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

Definition

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

  1. It has a subgroup of finite index that is nilpotent.
  2. It has a normal subgroup of finite index that is nilpotent.
  3. There is a surjective homomorphism from it to a finite group such that the kernel is a nilpotent normal subgroup.
  4. 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

Relation with other properties

Stronger properties

Weaker properties