# Finite nilpotent group

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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:

### 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 Finite Lazard Lie group, Finite group that is 1-isomorphic to an abelian group, Finite group that is order statistics-equivalent to an 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 Finite supersolvable group, Group having a Sylow tower, Group having subgroups of all orders dividing the group order|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