# Finite nilpotent group

From Groupprops

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

## Contents

## Definition

### Main equivalent definitions

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

- It is a nilpotent group
- It satisfies the normalizer condition i.e. it has no proper self-normalizing subgroup
- Every maximal subgroup is normal
- All its Sylow subgroups are normal
- It is the direct product of its Sylow subgroups
- It is a p-nilpotent group for every prime number (it suffices to check this condition only for those primes that divide the order). -nilpotent means that there exists a normal p-complement.
- It has a normal subgroup for every possible order dividing the group order
- 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 groupVIEW: 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 is a finite nilpotent group and is a subgroup of . Then, is also a finite nilpotent group. |

quotient-closed group property | Yes | follows from nilpotency is quotient-closed | Suppose is a finite nilpotent group and is a normal subgroup of . Then, is also a finite nilpotent group. |

finite direct product-closed group property | Yes | follows from nilpotency is finite direct product-closed | Suppose are finite nilpotent groups. Then, the external direct product 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 with isomorphic lattices of subgroups, such that is finite nilpotent and 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 ) | finite group such that there exists a normal -complement. | |FULL LIST, MORE INFO |