Virtually nilpotent group: Difference between revisions

Latest revision as of 09:06, 14 February 2012

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

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

Relation with other properties

Stronger properties

Virtually abelian group

Weaker properties

Virtually solvable group

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 ==Definition==
-A [[group]] is termed '''virtually nilpotent''' if it has a [[subgroup of finite index]] that is [[defining ingredient::nilpotent group|nilpotent]].
+A [[group]] is termed '''virtually nilpotent''' if it satisfies the following equivalent conditions:
+# It has a [[subgroup of finite index]] that is [[defining ingredient::nilpotent group|nilpotent]].
+# It has a [[normal subgroup of finite index]] that is [[nilpotent group|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==