Nilpotent group

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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VIEW RELATED: Group property implications | Group property non-implications | Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions |Group property dissatisfactions<random> @@@
RANDOM GROUP PROPERTY: <random>Simple group: A nontrivial group having no proper nontrivial normal subgroup.@@@Noetherian group: A group where every subgroup is finitely generated.@@@Complete group: A group for which the natural homomorphism to its automorphism group is an isomorphism, i.e., a centerless group for which every automorphism is inner.@@@SQ-universal group: A group for which every finitely generated group can be realized as a subquotient.@@@T-group: A group where any subnormal subgroup is normal, i.e., where normality is transitive. In general, normality is not transitive.@@@Locally finite group: A group in which every finitely generated subgroup is finite.@@@Group satisfying normalizer condition: A group with no proper self-normalizing subgroup.</random></random>

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 nilpotency class (sometimes written as nilpotence class) of the group. For any $c$ greater than or equal to than the nilpotency 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 nonnegative 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 the center $Z(G/Z^{c-1}(G))$ under the natural quotient map from $G$ to $G/Z^{c-1}(G)$, and $Z^{0}(G)$ is the trivial subgroup.

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

• There is a nonnegative integer $c$ such that $[[[..[G,G],G],G],...G]$ is trivial where $G$ is repeated $c+1$ times. Here, $[,]$ denotes the commutator of two subgroups. In other words, the lower central series of $G$ reaches the identity in finitely many steps.

• There is a nonnegative integer $c$ and a chain of subgroups:

$G=H_{1}\geq H_{2}\geq \dots \geq H_{c+1}=\{e\}$

such that each $H_{i}$ is a normal subgroup of $G$ and $H_{i}/H_{i+1}$ is in the center of $G/H_{i+1}$. In other words, there exists a central series for $G$ of length $c$.

The smallest possible $c$ for all three definitions is termed the nilpotency class (sometimes written nilpotence class) of $G$. We usually say a group is of nilpotency class $c$ if its nilpotency class is less than or equal to $c$.

Equivalence of definitions

Further information: Equivalence of definitions of nilpotent group, equivalence of definitions of nilpotency class

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 other properties obtained by applying the diagonal-in-square operator

A group $G$ is nilpotent if and only if the diagonal subgroup $\{(g,g)\mid g\in G\}$ is subnormal in the group $G\times G$. In fact, the nilpotency class of $G$ equals the subnormal depth of the diagonal subgroup.

Examples

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

• The trivial group is nilpotent, of nilpotency class zero.
• Any abelian group is nilpotent, of nilpotency class one.
• Any group of prime power order is nilpotent. Further information: prime power order implies nilpotent
• The dihedral group of order 8 is the smallest (in terms of order) nilpotent group which is not abelian. It is a group of nilpotency class two.
• The quaternion group is also the smallest (in terms of order) nilpotent group which is not abelian. This also has order eight.
• For $p\neq 2$, there are in general two non-abelian groups of order $p^{3}$, both of which are nilpotent (of class two): semidirect product of cyclic group of prime-square order and cyclic group of prime order and prime-cube order group:U(3,p).

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

• Variations of nilpotent group

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:

• Nilpotent normal subgroup
• Nilpotent characteristic subgroup
• Nilpotent subnormal subgroup

Stronger properties

property	meaning	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Abelian group	any two elements commute, so class $\leq 1$	abelian implies nilpotent	nilpotent not implies abelian (see also list of examples)	|FULL LIST, MORE INFO
Cyclic group	generated by one element	(via abelian)	(via abelian) (see also list of examples)	Abelian group|FULL LIST, MORE INFO
Group of prime power order	order is a power of a prime	prime power order implies nilpotent	(see also list of examples)	|FULL LIST, MORE INFO
Finite nilpotent group	Nilpotent and a finite group		(see also list of examples)	|FULL LIST, MORE INFO
Group of nilpotency class two	The commutator subgroup is in the center		(see also list of examples)	|FULL LIST, MORE INFO

Weaker properties

property	meaning	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Metanilpotent group	has nilpotent normal subgroup with nilpotent quotient		(see also list of examples)	|FULL LIST, MORE INFO
Solvable group	derived series reaches the trivial subgroup	nilpotent implies solvable	solvable not implies nilpotent (see also list of examples)	|FULL LIST, MORE INFO
Group in which every maximal subgroup is normal	every maximal subgroup is a normal subgroup	nilpotent implies every maximal subgroup is normal		|FULL LIST, MORE INFO
Group in which every subgroup is subnormal	every subgroup is a subnormal subgroup	nilpotent implies every subgroup is subnormal		|FULL LIST, MORE INFO
Group satisfying normalizer condition	no proper self-normalizing subgroup	nilpotent implies normalizer condition	normalizer condition not implies nilpotent	|FULL LIST, MORE INFO
Gruenberg group
Hypercentral group	transfinite upper central series terminates at whole group			|FULL LIST, MORE INFO
Hypocentral group	transfinite lower central series terminates at trivial subgroup			|FULL LIST, MORE INFO
Locally nilpotent group	every finitely generated subgroup is nilpotent			|FULL LIST, MORE INFO
Residually nilpotent group	for every element, there is a normal subgroup with nilpotent quotient not containing it			|FULL LIST, MORE INFO

Facts

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

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

For more specific kinds of facts:

• Category:Subgroup property implications in nilpotent groups
• Category:Subgroup metaproperty satisfactions in nilpotent groups

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 $d\geq c$. 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 of fixed class is quasivarietal, Nilpotence is quasivarietal

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
View a complete list of 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 a complete list of 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 finite direct product-closed group properties

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. Further information: Nilpotence is finite-normal join-closed

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, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{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

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• Mathworld page