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 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 ![]() ![]() ![]() ![]() |
quotient-closed group property | Yes | follows from nilpotency is quotient-closed | Suppose ![]() ![]() ![]() ![]() |
finite direct product-closed group property | Yes | follows from nilpotency is finite direct product-closed | Suppose ![]() ![]() |
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 ![]() ![]() ![]() |
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 ![]() |
|FULL LIST, MORE INFO |