Nilpotent normal subgroup

This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property (itself viewed as a subgroup property): nilpotent group
View a complete list of such conjunctions

Definition

A subgroup of a group is termed a nilpotent normal subgroup if it is nilpotent as a group, and normal as a subgroup.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
cyclic normal subgroup	cyclic as a group and normal as a subgroup	(via abelian normal)	(via abelian normal)	Abelian normal subgroup|FULL LIST, MORE INFO
elementary abelian normal subgroup	elementary abelian as a group and normal as a subgroup	(via abelian)	Abelian normal subgroup|FULL LIST, MORE INFO
central subgroup	subgroup contained in the center	(via abelian normal)	(via abelian normal)	Abelian normal subgroup|FULL LIST, MORE INFO
abelian normal subgroup	abelian as a group and normal as a subgroup	abelian implies nilpotent	nilpotent not implies abelian	|FULL LIST, MORE INFO
Dedekind normal subgroup	Dedekind group and normal as a subgroup (Dedekind means every subgroup is normal in it)	Dedekind implies nilpotent	nilpotent not implies Dedekind	|FULL LIST, MORE INFO
join of finitely many abelian normal subgroups
cyclic characteristic subgroup		(via cyclic normal)	(via cyclic normal)	|FULL LIST, MORE INFO
abelian characteristic subgroup		(via abelian normal)	(via abelian normal)	Abelian normal subgroup|FULL LIST, MORE INFO
nilpotent characteristic subgroup		characteristic implies normal	every nontrivial normal subgroup is potentially normal-and-not-characteristic, or normal not implies characteristic in any nontrivial subquasivariety of the quasivariety of groups	|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
solvable normal subgroup
normal subgroup

Facts

The subgroup of a group generated by all its nilpotent normal subgroups is termed the Fitting subgroup, and if a group equals its Fitting subgroup, then it is termed a Fitting group. For finite groups, the Fitting subgroup is the largest nilpotent normal subgroup, and a Fitting group is the same thing as a nilpotent group.