Nilpotent normal subgroup

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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, Dedekind normal subgroup, Homocyclic 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, Class two normal subgroup, Dedekind 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 Class two normal subgroup, Dedekind normal subgroup|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) Cyclic normal subgroup|FULL LIST, MORE INFO
abelian characteristic subgroup (via abelian normal) (via abelian normal) Abelian normal subgroup, Nilpotent characteristic 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.