Nilpotent group explained

In mathematics, specifically group theory, a nilpotent group G is a group that has an upper central series that terminates with G. Equivalently, it has a central series of finite length or its lower central series terminates with .

Intuitively, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute. It is also true that finite nilpotent groups are supersolvable. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.^[1]

Nilpotent groups arise in Galois theory, as well as in the classification of groups. They also appear prominently in the classification of Lie groups.

Analogous terms are used for Lie algebras (using the Lie bracket) including nilpotent, lower central series, and upper central series.

Definition

The definition uses the idea of a central series for a group. The following are equivalent definitions for a nilpotent group :

For a nilpotent group, the smallest such that has a central series of length is called the nilpotency class of ; and is said to be nilpotent of class . (By definition, the length is if there are

n+1

different subgroups in the series, including the trivial subgroup and the whole group.)

Equivalently, the nilpotency class of equals the length of the lower central series or upper central series.If a group has nilpotency class at most, then it is sometimes called a nil- group.

It follows immediately from any of the above forms of the definition of nilpotency, that the trivial group is the unique group of nilpotency class , and groups of nilpotency class are exactly the non-trivial abelian groups.^[2]

Examples

As noted above, every abelian group is nilpotent.^[3] ^[4]
For a small non-abelian example, consider the quaternion group Q₈, which is a smallest non-abelian p-group. It has center of order 2, and its upper central series is,, Q₈; so it is nilpotent of class 2.
The direct product of two nilpotent groups is nilpotent.^[5]
All finite p-groups are in fact nilpotent (proof). For n > 1, the maximal nilpotency class of a group of order pⁿ is n - 1 (for example, a group of order p² is abelian). The 2-groups of maximal class are the generalised quaternion groups, the dihedral groups, and the semidihedral groups.
Furthermore, every finite nilpotent group is the direct product of p-groups.^[5]
The multiplicative group of upper unitriangular n × n matrices over any field F is a nilpotent group of nilpotency class n − 1. In particular, taking n = 3 yields the Heisenberg group H, an example of a non-abelian^[6] infinite nilpotent group.^[7] It has nilpotency class 2 with central series 1, Z(H), H.
The multiplicative group of invertible upper triangular n × n matrices over a field F is not in general nilpotent, but is solvable.
Any nonabelian group G such that G/Z(G) is abelian has nilpotency class 2, with central series, Z(G), G.

The natural numbers k for which any group of order k is nilpotent have been characterized .

Explanation of term

Nilpotent groups are called so because the "adjoint action" of any element is nilpotent, meaning that for a nilpotent group

of nilpotence degree

and an element

, the function

\operatorname{ad}_g\colonG\toG

defined by

\operatorname{ad}_g(x):=[g,x]

(where

[g,x]=g^-1x^-1gx

is the commutator of

and

) is nilpotent in the sense that the

th iteration of the function is trivial:

	n(x)=e
\left(\operatorname{ad}
	g\right)

for all

This is not a defining characteristic of nilpotent groups: groups for which

\operatorname{ad}_g

is nilpotent of degree

(in the sense above) are called

-Engel groups,^[8] and need not be nilpotent in general. They are proven to be nilpotent if they have finite order, and are conjectured to be nilpotent as long as they are finitely generated.

An abelian group is precisely one for which the adjoint action is not just nilpotent but trivial (a 1-Engel group).

Properties

Since each successive factor group in the upper central series is abelian, and the series is finite, every nilpotent group is a solvable group with a relatively simple structure.

Every subgroup of a nilpotent group of class is nilpotent of class at most ;^[9] in addition, if is a homomorphism of a nilpotent group of class, then the image of is nilpotent of class at most .

Statement (d) can be extended to infinite groups: if is a nilpotent group, then every Sylow subgroup of is normal, and the direct product of these Sylow subgroups is the subgroup of all elements of finite order in (see torsion subgroup).

Many properties of nilpotent groups are shared by hypercentral groups.

References

Book: Bechtell, Homer . The Theory of Groups . . 1971 .
Book: Friedrich . Von Haeseler . Automatic Sequences . De Gruyter Expositions in Mathematics . 36 . . Berlin . 2002 . 3-11-015629-6 .
Book: Hungerford . Thomas W. . Thomas W. Hungerford . Algebra . Springer-Verlag . 1974 . 0-387-90518-9 .
Book: Isaacs, I. Martin . Martin Isaacs. Finite Group Theory. 2008. American Mathematical Society. 978-0-8218-4344-4.
Book: Palmer, Theodore W. . Banach Algebras and the General Theory of *-algebras . . 1994 . 0-521-36638-0 .
Book: Urs . Stammbach . Homology in Group Theory . Lecture Notes in Mathematics . 359 . Springer-Verlag . 1973 . review
Book: Suprunenko, D. A. . Matrix Groups . . Providence, Rhode Island . 1976 . 0-8218-1341-2 .
Book: Tabachnikova, Olga . Smith, Geoff . Topics in Group Theory . Springer Undergraduate Mathematics Series . Springer . 2000 . 1-85233-235-2 .
Book: Zassenhaus . Hans . Hans Zassenhaus . The Theory of Groups . . New York . 1999 . 0-486-40922-8 .

Notes and References

Dixon. M. R.. Kirichenko. V. V.. Kurdachenko. L. A.. Otal. J.. Semko. N. N.. Shemetkov. L. A.. Subbotin. I. Ya.. S. N. Chernikov and the development of infinite group theory. Algebra and Discrete Mathematics. 2012. 13. 2. 169–208.
Book: Tabachnikova & Smith . [{{Google books|plainurl=y|id=DD0TW28WjfQC|page=169|text=The trivial group has nilpotency class 0}} Topics in Group Theory (Springer Undergraduate Mathematics Series)]. 2000. 169.
Book: Suprunenko . [{{Google books|plainurl=y|id=cTtuPOj5h10C|page=205|text=abelian group is nilpotent}} Matrix Groups]. 1976. 205.
Book: Hungerford . [{{Google books|plainurl=y|id=t6N_tOQhafoC|page=100|text=every abelian group G is nilpotent}} Algebra]. 1974. 100.
Book: Zassenhaus . [{{Google books|plainurl=y|id=eCBK6tj7_vAC|page=143|text=The direct product of a finite number of nilpotent groups is nilpotent}} The theory of groups]. 1999. 143.
Book: Haeseler . [{{Google books|plainurl=y|id=wmh7tc6uGosC|page=15|text=The Heisenberg group is a non-abelian}} Automatic Sequences (De Gruyter Expositions in Mathematics, 36)]. 2002. 15.
Book: Palmer . [{{Google books|plainurl=y|id=zn-iZNNTb-AC|page=1283|text=Heisenberg group this group has nilpotent length 2 but is not abelian}} Banach algebras and the general theory of *-algebras]. 2001. 1283.
For the term, compare Engel's theorem, also on nilpotency.
Bechtell (1971), p. 51, Theorem 5.1.3