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 subgroupFULL LIST, MORE INFO

elementary abelian normal subgroup 
elementary abelian as a group and normal as a subgroup 
(via abelian) 
Abelian normal subgroupFULL 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 subgroupFULL 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 subgroupFULL 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 subgroupFULL LIST, MORE INFO

abelian characteristic subgroup 

(via abelian normal) 
(via abelian normal) 
Abelian normal subgroup, Nilpotent characteristic subgroupFULL LIST, MORE INFO

nilpotent characteristic subgroup 

characteristic implies normal 
every nontrivial normal subgroup is potentially normalandnotcharacteristic, or normal not implies characteristic in any nontrivial subquasivariety of the quasivariety of groups 
FULL LIST, MORE INFO

Weaker properties
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.