Nilpotent group

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This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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This article defines a group property that is pivotal (i.e., important) among existing group properties
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RANDOM GROUP PROPERTY: Locally finite group: A group in which every finitely generated subgroup is finite.

The version of this for finite groups is at: finite nilpotent group

Definition

Symbol-free definition

A group is said to be nilpotent if it satisfies the following equivalent conditions:

Its upper central series stabilizes after a finite length at the whole group.
Its lower central series stabilizes after a finite length at the trivial subgroup.
It possesses a central series.

The length after which the upper central series stabilizes equals the length after which the lower central series stabilizes, and this length is termed the nilpotence class of the group. For any $c$ greater than or equal to than the nilpotence class, the group is said to be of class $c$

Definition with symbols

A group $G$ is said to be nilpotent if it satisfies the following equivalent conditions:

There is a positive integer $c$ such that $Z c (G) = G$ . Here, we define $Z c (G)$ inductively as follows:

$Z c (G)$ is the inverse image of $Z (G / Z c - 1 (G))$ under the natural projection from $G$ to $G / Z (G)$ .

The subgroups $Z c (G)$ are said to form the upper central series of $G$ .

There is a positive integer $c$ such that $[[[..[G, G], G], G],... G]$ is trivial where $G$ is repeated $c + 1$ times. In other words, the lower central series reaches the identity in finitely many steps.

The smallest $c$ for both is termed the nilpotence class of $G$ . We sometimes say a group is of nilpotence class $c$ if its nilpotence class is less than or equal to $c$ .

Equivalence of definitions

Further information: Equivalence of definitions of nilpotent group

Formalisms

In terms of ascending series

This group property is obtained by applying the ascending series-finite operator to the subgroup-defining function: center

A group is nilpotent if and only if the ascending series corresponding to the center subgroup-defining function (which is the upper central series) terminates at the whole group in finitely many steps.

In terms of the diagonal-in-square operator

This property is obtained by applying the diagonal-in-square operator to the property: subnormal subgroup
View all properties obtained by applying the diagonal-in-square operator

A group $G$ is nilpotent if and only if the diagonal subgroup is subnormal in $G \times G$ . In fact, the nilpotence class of $G$ equals the subnormal depth of the diagonal.

Examples

A full list of nilpotent groups we encounter is at Category:Nilpotent groups. Some important examples are given below:

The dihedral group of order 8 is the smallest (in terms of order) nilpotent group which is not Abelian
The quaternion group is also the smallest (in terms of order) nilpotent group which is not Abelian
For $p \ne 2$ , there are in general two non-Abelian groups of order $p 3$ , both of which are nilpotent

Relation with other properties

This property is a pivotal (important) member of its property space. Its variations, opposites, and other properties related to it and defined using it are often studied

Category:Variations of nilpotence

Conjunction with other properties

Conjunctions with other group properties:

Finite nilpotent group: Conjunction of being finite and nilpotent. Any finite nilpotent group is a direct product of its Sylow subgroups.
Finitely generated nilpotent group
Periodic nilpotent group

Conjunctions with subgroup properties:

Stronger properties

Abelian group: For full proof, refer: Abelian implies nilpotent
Group of nilpotence class two
Group of prime power order

Weaker properties

Metanilpotent group
Solvable group: For proof of the implication, refer Nilpotent implies solvable and for proof of its strictness (i.e. the reverse implication being false) refer Solvable not implies nilpotent.
Group in which every maximal subgroup is normal: For proof of the implication, refer Nilpotent implies every maximal subgroup is normal and for proof of its strictness (i.e. the reverse implication being false) refer Every maximal subgroup is normal not implies nilpotent.
Group in which every subgroup is subnormal
Group satisfying normalizer condition: For proof of the implication, refer Nilpotent implies normalizer condition and for proof of its strictness (i.e. the reverse implication being false) refer Normalizer condition not implies nilpotent.
Gruenberg group
Hypercentral group
Hypocentral group
Locally nilpotent group
Residually nilpotent group

Facts

A complete list of facts about nilpotent groups is available at:

Special:SearchByProperty/Fact-20about/Nilpotent-20group

For more specific kinds of facts:

Metaproperties

Quasivarietal group property

This group property is varietal, in the sense that the collection of groups satisfying this property forms a quasivariety of algebras. In other words, the collection of groups satisfying this property is closed under taking subgroups, taking quotients and taking finite direct products
View other quasivarietal group properties

The property of being nilpotent of class $c$ , for any fixed $c$ is varietal, and we further have that any group of nilpotence class $c$ is of nilpotence class $d$ for any $d \ge c$ . Combining these two facts, we obtain that:

Any subgroup of a nilpotent group is nilpotent. In fact, any subgroup of a group of nilpotence class $c$ has nilpotence class $c$ .
Any quotient of a nilpotent group is nilpotent. In fact, any quotient of a group of nilpotence class $c$ has nilpotence class $c$ .
Any direct product of two nilpotent groups is nilpotent. In fact, if both of then are of nilpotence class $c$ (we can take $c$ as the higher of their nilpotence classes) then their product is also of nilpotence class $c$ .

For full proof, refer: Nilpotence is quasivarietal

Subgroups

This group property is subgroup-closed, viz any subgroup of a group satisfying the property also satisfies the property
View other subgroup-closed group properties

Nilpotence is subgroup-closed on account of being quasivarietal. See above.

Quotients

This group property is quotient-closed, viz any quotient of a group satisfying the property also has the property
View other quotient-closed group properties

Nilpotence is quotient-closed on account of being quasivarietal. See above.

Direct products

This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property
View other group properties closed under finite direct products

Nilpotence is closed under finite direct products, on account of being quasivarietal. See above.

Finite normal joins

This group property is finite normal join-closed: in other words, a join of finitely many normal subgroups each having the group property, also has the group property

Nilpotence is closed under taking joins of finitely many normal subgroups. In other words, if a group is generated by finitely many nilpotent normal subgroups, it is also nilpotent.

Testing

GAP command

This group property can be tested using built-in functionality of Groups, Algorithms, Programming (GAP).
The GAP command for this group property is:IsNilpotentGroup
The class of all groups with this property can be referred to with the built-in command: NilpotentGroups
View GAP-testable group properties

To test whether a given group is nilpotent or not using GAP, enter:

IsNilpotentGroup (group);

where group is either the definition of a group or a name for a group already defined.

The class of all nilpotent groups is specified as NilpotentGroups.

Study of this notion

Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20F18

While 20F18 is the subject class used for nilpotent groups, the subject class used for finite nilpotent groups in particular is 20D15.

Closely related is 20F19: Generalizations of nilpotent and solvable groups.

References

Textbook references

Abstract Algebra by David S. Dummit and Richard M. Foote, ISBN 0471433349, ^{More info}, Page 190 (formal definition, along with lower central series and upper central series)
Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261, ^{More info}, Page 103 (definition introduced in paragraph, along with lower central series and upper central series)
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 122 (formal definition, in terms of central series)
An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444, ^{More info}, Page 174 (formal definition, in terms of central series)
Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189, ^{More info}, Page 100 (definition introduced in paragraph)
Topics in Algebra by I. N. Herstein, ^{More info}, Page 117 (definition introduced based on exercises 13-14, that implicitly define lower central series and upper central series, and precedes exercise 15)

External links

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Definition links

Facts about Nilpotent groupRDF feed

Defined in	DummitFoote (?, ?, ?) +, AlperinBell (?, ?, ?) +, RobinsonGT (?, ?, ?) +, RobinsonAA (?, ?, ?) +, Hungerford (?, ?, ?) +, Herstein (?, ?, ?) +, Wikipedia (?, ?, ?) +, Planetmath (?, ?, ?) +, and Mathworld (?, ?, ?) +
Defining ingredient	Diagonal-in-square operator +, and Subnormal subgroup +
MSC class	20F18 +
Referenced in	DummitFoote (?, ?, ?) +, AlperinBell (?, ?, ?) +, RobinsonGT (?, ?, ?) +, RobinsonAA (?, ?, ?) +, Hungerford (?, ?, ?) +, Herstein (?, ?, ?) +, Wikipedia (?, ?, ?) +, Planetmath (?, ?, ?) +, and Mathworld (?, ?, ?) +
Stronger than	Metanilpotent group +, Solvable group +, Group in which every maximal subgroup is normal +, Group in which every subgroup is subnormal +, Group satisfying normalizer condition +, Gruenberg group +, Hypercentral group +, Hypocentral group +, Locally nilpotent group +, and Residually nilpotent group +
Weaker than	Abelian group +, Group of nilpotence class two +, and Group of prime power order +