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
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.
