# Nilpotent group

﻿
Nilpotent group
Types of groups Concepts in group theory category of groups subgroups, normal subgroups group homomorphisms, kernel, image, quotient direct product, direct sum semidirect product, wreath product simple, finite, infinite discrete, continuous multiplicative, additive cyclic, abelian, dihedral nilpotent, solvable list of group theory topics glossary of group theory

In mathematics, more specifically in the field of group theory, 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.

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, explained on its own page, of a central series for a group. The following are equivalent formulations:

• A nilpotent group is one that has a central series of finite length.
• A nilpotent group is one whose lower central series terminates in the trivial subgroup after finitely many steps.
• A nilpotent group is one whose upper central series terminates in the whole group after finitely many steps.

For a nilpotent group, the smallest n such that G has a central series of length n is called the nilpotency class of G ; and G is said to be nilpotent of class n. (By definition, the length is n if there are n + 1 different subgroups in the series, including the trivial subgroup and the whole group.)

Equivalently, the nilpotency class of G equals the length of the lower central series or upper central series. If a group has nilpotency class at most m, then it is sometimes called a nil-m 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 0, and groups of nilpotency class 1 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 Q8, which is a smallest non-abelian p-group. It has center {1, −1} of order 2, and its upper central series is {1}, {1, −1}, Q8; so it is nilpotent of class 2.
• All finite p-groups are in fact nilpotent (proof). The maximal class of a group of order pn is n - 1. The 2-groups of maximal class are the generalised quaternion groups, the dihedral groups, and the semidihedral groups.
• The direct product of two nilpotent groups is nilpotent.
• Conversely, every finite nilpotent group is the direct product of p-groups.
• The Heisenberg group is an example of non-abelian, infinite nilpotent group.
• The multiplicative group of upper unitriangular n x n matrices over any field F is a nilpotent group of nilpotent length n - 1 .
• The multiplicative group of invertible upper triangular n x n matrices over a field F is not in general nilpotent, but is solvable.

## Explanation of term

Nilpotent groups are so called because the "adjoint action" of any element is nilpotent, meaning that for a nilpotent group G of nilpotence degree n and an element g, the function $\operatorname{ad}_g \colon G \to G$ defined by $\operatorname{ad}_g(x) := [g,x]$ (where [g,x] = g − 1x − 1gx is the commutator of g and x) is nilpotent in the sense that the nth iteration of the function is trivial: $\left(\operatorname{ad}_g\right)^n(x)=e$ for all x in G.

This is not a defining characteristic of nilpotent groups: groups for which $\operatorname{ad}_g$ is nilpotent of degree n (in the sense above) are called n-Engel groups, 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 Zi+1/Zi 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 n is nilpotent of class at most n; in addition, if f is a homomorphism of a nilpotent group of class n, then the image of f is nilpotent of class at most n.

The following statements are equivalent for finite groups, revealing some useful properties of nilpotency:

• G is a nilpotent group.
• If H is a proper subgroup of G, then H is a proper normal subgroup of NG(H) (the normalizer of H in G). This is called the normalizer property and can be phrased simply as "normalizers grow".
• Every maximal proper subgroup of G is normal.
• G is the direct product of its Sylow subgroups.

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

Many properties of nilpotent groups are shared by hypercentral groups.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Nilpotent — This article is about a type of element in a ring. For the type of group, see Nilpotent group. In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. The term was… …   Wikipedia

• Nilpotent orbit — Nilpotent orbits are generalizations of nilpotent matrices that play an important role in representation theory of real and complex semisimple Lie groups and semisimple Lie algebras. Contents 1 Definition 2 Examples 3 Properties …   Wikipedia

• Group scheme — In mathematics, a group scheme is a group object in the category of schemes. That is, it is a scheme G with the equivalent properties* there is a group law expressible as a multiplication μ and inversion map ι on G ; or * G is a functor (as in… …   Wikipedia

• Core (group) — In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the p core of a group. Contents 1 The normal core 1.1 Definition 1.2… …   Wikipedia

• Solvable group — Concepts in group theory category of groups subgroups, normal subgroups group homomorphisms, kernel, image, quotient direct product, direct sum semidirect product, wreath product …   Wikipedia

• Heisenberg group — In mathematics, the Heisenberg group, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form or its generalizations under the operation of matrix multiplication. Elements a, b, c can be taken from some… …   Wikipedia

• Engel group — In mathematics, an element x of a Lie group or a Lie algebra is called an n Engel element, named after Friedrich Engel, if it satisfies the n Engel condition that the repeated commutator [ x , y ] , y ] , ..., y ] with n copies of y is trivial… …   Wikipedia

• p-group — Not to be confused with n group. In mathematics, given a prime number p, a p group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there… …   Wikipedia

• A-group — In mathematics, in the area of abstract algebra known as group theory, an A group is a type of group that is similar to abelian groups. The groups were first studied in the 1940s by Philip Hall, and are still studied today. A great deal is known… …   Wikipedia

• P-group — In mathematics, given a prime number p , a p group is a periodic group in which each element has a power of p as its order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the …   Wikipedia