Finite nilpotent group

This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
View other properties of finite groups OR View all group properties

Definition

Main equivalent definitions

A finite group is termed a finite nilpotent group if it satisfies the following equivalent conditions:

1. It is a nilpotent group.
2. It satisfies the normalizer condition i.e. it has no proper self-normalizing subgroup.
3. Every maximal subgroup is normal.
4. All its Sylow subgroups are normal.
5. It is the direct product of its Sylow subgroups.
6. It is a p-nilpotent group for every prime number $p$ (it suffices to check this condition only for those primes that divide the order). $p$-nilpotent means that there exists a normal p-complement.
7. It has a normal subgroup for every possible order dividing the group order.
8. Every normal subgroup of the group contains a normal subgroup of the group for every order dividing the order of the normal subgroup.

Other equivalent definitions that are weaker versions of nilpotent in the general case

The following is a list of group properties, each weaker than being nilpotent, that for a finite group turn out to be equivalent to being nilpotent:

• Group satisfying normalizer condition: It has no proper self-normalizing subgroup
• Group in which every subgroup is subnormal
• Locally nilpotent group
• Residually nilpotent group
• Engel group: See Zorn's theorem on Engel groups
• Hypercentral group
• Hypocentral group

Equivalence of definitions

Further information: Equivalence of definitions of finite nilpotent group

Examples

VIEW: groups satisfying this property | groups dissatisfying property finite group | groups dissatisfying property nilpotent group
VIEW: Related group property satisfactions | Related group property dissatisfactions

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	Yes	follows from nilpotency is subgroup-closed	Suppose $G$ is a finite nilpotent group and $H$ is a subgroup of $G$. Then, $H$ is also a finite nilpotent group.
quotient-closed group property	Yes	follows from nilpotency is quotient-closed	Suppose $G$ is a finite nilpotent group and $H$ is a normal subgroup of $G$. Then, $G/H$ is also a finite nilpotent group.
finite direct product-closed group property	Yes	follows from nilpotency is finite direct product-closed	Suppose $G_{1},G_{2},\dots ,G_{n}$ are finite nilpotent groups. Then, the external direct product $G_{1}\times G_{2}\times \dots \times G_{n}$ is also a finite nilpotent group.
lattice-determined group property	No	there exists an abelian group of prime power order that is lattice-isomorphic to a non-abelian group not of prime power order	It is possible to have groups $G_{1},G_{2}$ with isomorphic lattices of subgroups, such that $G_{1}$ is finite nilpotent and $G_{2}$ is not.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite abelian group				|FULL LIST, MORE INFO
group of prime power order				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite solvable group				|FULL LIST, MORE INFO
finite supersolvable group				|FULL LIST, MORE INFO
periodic nilpotent group	nilpotent and periodic: every element has finite order			|FULL LIST, MORE INFO
locally finite nilpotent group	nilpotent and locally finite: every finitely generated subgroup is finite			|FULL LIST, MORE INFO
finitely generated nilpotent group	nilpotent and finitely generated			|FULL LIST, MORE INFO
p-nilpotent group (for any fixed prime number $p$)	finite group such that there exists a normal $p$-complement.			|FULL LIST, MORE INFO