In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
, a nilpotent group ''G'' is a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
that has an
upper central series that terminates with ''G''. Equivalently, its
central series is of finite length or its
lower central series
In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a centra ...
terminates with .
Intuitively, a nilpotent group is a group that is "almost
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
". This idea is motivated by the fact that nilpotent groups are
solvable, and for finite nilpotent groups, two elements having
relatively prime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
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.
Nilpotent groups arise in
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
, as well as in the classification of groups. They also appear prominently in the classification of
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s.
Analogous terms are used for
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s (using the
Lie bracket) including
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cl ...
, 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
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.
Examples
* As noted above, every abelian group is nilpotent.
* For a small non-abelian example, consider the
quaternion group
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset
\ of the quaternions under multiplication. It is given by the group presentation
:\mathrm_8 ...
''Q''
8, which is a smallest non-abelian ''p''-group. It has center of order 2, and its upper central series is , , ''Q''
8; so it is nilpotent of class 2.
* The
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
of two nilpotent groups is nilpotent.
* All finite
''p''-groups are in fact nilpotent (
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a c ...
). The maximal class of a group of order ''p''
''n'' is ''n'' (for example, any group of order 2 is nilpotent of class 1). The 2-groups of maximal class are the generalised
quaternion group
In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset
\ of the quaternions under multiplication. It is given by the group presentation
:\mathrm_8 ...
s, the
dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
s, and the
semidihedral groups.
* Furthermore, every finite nilpotent group is the direct product of ''p''-groups.
* 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
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
::\begin
1 & a & c\\
0 & 1 & b\\
0 & 0 & 1\\
\end
under the operation of matrix multiplication. Elements ...
''H'', an example of a non-abelian infinite nilpotent group. 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 so called because the "adjoint action" of any element is
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cl ...
, meaning that for a nilpotent group
of nilpotence degree
and an element
, the function
defined by