Nilpotent group

Definition

Equivalent definitions in tabular format

No.	Shorthand	A group is termed nilpotent if ...	A group $G$ is termed nilpotent if ...
1	upper central series	its upper central series stabilizes after a finite length at the whole group	there is a nonnegative integer $c$ such that $Z^c(G) = G$. Here, we define $Z^i(G)$ inductively as follows: $Z^i(G)$ is the inverse image of the center $Z(G/Z^{i-1}(G))$ under the natural quotient map from $G$ to $G/Z^{i-1}(G)$, and $Z^0(G)$ is the trivial subgroup.
2	lower central series	its lower central series stabilizes after a finite length at the trivial subgroup	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.
3	central series	it possesses a central series	there is a nonnegative integer $c$ and a chain of subgroups: $G = H_1 \ge H_2 \ge \dots \ge H_{c+1} = \{ e \}$ 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 $c$.
4	diagonal subnormal in square	the diagonal subgroup is subnormal in the square of the group	the subgroup $\{ (g,g) : g \in G \}$ is subnormal in the square $G \times G$ with subnormal depth $c$.
5	iterated left-normed commutators trivial	there is a finite length such that any iterated left-normed commutator of length more than that becomes trivial	there is a length $c$ such that any commutator of the form $[[\dots[[x_1,x_2],x_3],\dots],x_{c+1}]$ takes value the identity element, where the $x_i$ are (possibly repeated) elements of $G$.
6	iterated commutators of any form trivial	there is a finite length such that any iterated commutator (with any kind of parenthesization of terms) of length more than that becomes trivial	there is a length $c$ such that any iterated commutator that involves at least $c$ commutator operations (so $c+1$ original inputs) takes value the identity element. [SHOW MORE] For instance, for $c = 3$, the expressions $\! [[x_1,x_2],[x_3,x_4]], [[[x_1,x_2],x_3],x_4]], [x_1,[x_2,[x_3,x_4]]]$, $[[x_1,[x_2,x_3]],x_4], [x_1,[[x_2,x_3],x_4]$ all take value the identity element.
7	iterated left-normed commutators trivial (generating set version)	(pick a generating set for the group) there is a finite length such that any iterated left-normed commutator of elements from that generating set length more than that becomes trivial	pick a generating set $S$ for $G$. There is a length $c$ such that any commutator of the form $[[\dots[[x_1,x_2],x_3],\dots],x_{c+1}]$ takes value the identity element, where the $x_i$ are (possibly repeated) elements of $S$.
8	iterated commutators of any form trivial (generating set version)	(pick a generating set for the group) there is a finite length such that any iterated commutator (with any kind of parenthesization of terms) of elements from that generating set of length more than that becomes trivial	pick a generating set $S$ for $G$. There is a length $c$ such that any iterated commutator that involves at least $c$ commutator operations (so $c+1$ original inputs) and where the inputs are from $S$, always takes value the identity element. [SHOW MORE] For instance, for $c = 3$, the expressions $\! [[x_1,x_2],[x_3,x_4]], [[[x_1,x_2],x_3],x_4]], [x_1,[x_2,[x_3,x_4]]]$, $[[x_1,[x_2,x_3]],x_4], [x_1,[[x_2,x_3],x_4]$ all take value the identity element.

The smallest possible $c$ for all 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 $c$.

This definition is presented using a tabular format. |View all pages with definitions in tabular format

Equivalence of definitions

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

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 (note that the nilpotency class is exactly one for nontrivial abelian groups).
• Any group of prime power order is nilpotent. Further information: prime power order implies nilpotent. Thus, exploring groups of prime power order is a good starting point for exploring nilpotent groups. See groups of order 8, groups of order 16, groups of order 32, groups of order 27, groups of order 81. See also nilpotency class distribution of finite p-groups.
• 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 $p \ne 2$, 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 unitriangular matrix group:UT(3,p).

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
View a list of pivotal group properties | View a complete list of group properties [SHOW MORE]

VIEW RELATED: Group property implications | Group property non-implications | Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions |Group property dissatisfactions
RANDOM GROUP PROPERTY: Group satisfying normalizer condition: A group with no proper self-normalizing subgroup.

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

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
pseudovarietal group property	Yes	nilpotency is pseudovarietal	Nilpotency is closed under taking subgroups, quotient groups, and finite direct products.
subgroup-closed group property	Yes	nilpotency is subgroup-closed	If $G$ is nilpotent, and $H \le G$, then $H$ is nilpotent.
quotient-closed group property	Yes	nilpotency is quotient-closed	If $G$ is nilpotent, and $H \le G$ is a normal subgroup, the quotient group $G/H$ is nilpotent.
finite direct product-closed group property	Yes	nilpotency is finite direct product-closed	If $G_1,G_2, \dots, G_n$ are all nilpotent groups, the external direct product $G_1 \times G_2 \times \dots \times G_n$ is also nilpotent.
finite normal join-closed group property	Yes	nilpotency is finite normal join-closed	Suppose $G$ is a group and $N_1,N_2,\dots,N_r$ are all nilpotent normal subgroups of $G$. Then the join of subgroups (which in this case is also the product of subgroups) $N_1N_2 \dots N_r$ is also nilpotent.
extension-closed group property	No	nilpotency is not extension-closed	It is possible to have a non-nilpotent group $G$ and a normal subgroup $H$ such that both $H$ and $G/H$ are nilpotent.
isoclinism-invariant group property	Yes	isoclinic groups have same nilpotency class	If $G_1,G_2$ are isoclinic groups, then $G_1$ is nilpotent if and only if $G_2$ is nilpotent. Further, they have the same nilpotency class (except in the case where one of them is trivial and the other is nontrivial abelian).

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:

Conjunction	Other component of conjunction	Can it be verified using only the abelianization?	Alternative possibilities for other component of conjunction that give the same result	More about the conjunction
finite nilpotent group	finite group	Yes		There are many alternative definitions of finite nilpotent group. The most relevant is that a finite group is nilpotent if and only if it is the internal direct product of its Sylow subgroups.
finitely generated nilpotent group	finitely generated group	Yes	finitely presented group Noetherian group	The key fact is that the property of being finitely generated can be tested knowing the abelianization. This is because any set of coset representatives for a generating set for the abelianization generate the whole group.
periodic nilpotent group	periodic group	Yes	locally finite group	the abelianization suffices because each of the successive quotients for the lower central series are homomorphic images of tensor powers of the abelianization.
divisible nilpotent group (or more generally, nilpotent group that is divisible for a set of primes)	divisible group (or more generally, divisible group for a set of primes)	Yes
rationally powered nilpotent group (or more generally, nilpotent group that is powered over a set of primes)	rationally powered group (or more generally, powered group for a set of primes)	In one direction (if the group is powered, so is the abelianization) but not in the other (it is possible for the abelianization to be powered and for the group to not be powered)		See derived subgroup is quotient-powering-invariant in nilpotent group and also nilpotent group with rationally powered abelianization need not be rationally powered
torsion-free nilpotent group (or more generally, nilpotent group that is torsion-free for a set of primes)	torsion-free group (or more generally, torsion-free group for a set of primes)	No (neither direction)		See nilpotent group with torsion-free abelianization need not be torsion-free nilpotent and torsion-free not implies torsion-free abelianization

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 $\le 1$	abelian implies nilpotent	nilpotent not implies abelian (see also list of examples)	Group in which class equals maximum subnormal depth, Group of nilpotency class three, Group of nilpotency class two, Group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two, UL-equivalent group|FULL LIST, MORE INFO
cyclic group	generated by one element	(via abelian)	(via abelian) (see also list of examples)	Epinilpotent group, Group of nilpotency class two, Group whose automorphism group is nilpotent|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)	Finitely generated nilpotent group, Periodic nilpotent group|FULL LIST, MORE INFO
group of nilpotency class two	The derived subgroup is in the center		(see also list of examples)	Group of nilpotency class three|FULL LIST, MORE INFO
group whose automorphism group is nilpotent	The automorphism group is a nilpotent group	nilpotent automorphism group implies nilpotent of class at most one more	nilpotent not implies nilpotent automorphism group	|FULL LIST, MORE INFO
UL-equivalent group	The upper central series and lower central series coincide	(by definition)	nilpotent not implies UL-equivalent (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)	Metanilpotent group|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		Group in which every subgroup is subnormal, Group satisfying normalizer condition|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	Group in which every subgroup is subnormal|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			Baer group, Group satisfying normalizer condition, Gruenberg group|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
n-nilpotent group for an integer $n$	The n-lower central series, obtained by iteration of the n-commutator operation, reaches the trivial subgroup in finitely many steps.

Testing

The testing problem

Further information: fixed-class nilpotency testing problem

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.

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.

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

Book	Page number	Chapter and section	Contextual information	View
Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347More info	190		formal definition, along with lower central series and upper central series
Groups and representations by Jonathan Lazare Alperin and Rowen B. Bell, ISBN 0387945261More info	103		definition in paragraph, along with lower central series and upper central series
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613More info	122		formal definition, in terms of central series
An Introduction to Abstract Algebra by Derek J. S. Robinson, ISBN 3110175444More info	174		formal definition, in terms of central series
Algebra (Graduate Texts in Mathematics) by Thomas W. Hungerford, ISBN 0387905189More info	100		definition in paragraph
Topics in Algebra by I. N. HersteinMore info	117		definition introduced based on exercises 13-14, that implicitly define lower central series and upper central series, and precedes exercise 15