Nilpotent group - Citizendia

In group theory, a nilpotent group is a group having a special property that makes it "almost" abelian, through repeated application of the commutator operation, [x,y] = x^-1y^-1xy. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. Nilpotent groups arise in Galois theory, as well as in the classification of groups. In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory They also appear prominently in the classification of Lie groups. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group

Analogous terms are used for Lie algebras (using the Lie bracket) including nilpotent, lower central series, and upper central series. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie Lie bracket can refer to Lie algebra Lie bracket of vector fields

1 Definition
- 1.1 Explanation of term
2 Examples
3 Properties
4 References

Definition

The following are equivalent definitions for a nilpotent group:

A nilpotent group is one that has a central series of finite length. In Mathematics, especially in the fields of Group theory and Lie theory, a central series is a kind of Normal series of Subgroups or
A nilpotent group is one whose lower central series terminates in the trivial subgroup after finitely many steps. In Mathematics, especially in the fields of Group theory and Lie theory, a central series is a kind of Normal series of Subgroups or
A nilpotent group is one whose upper central series terminates in the whole group after finitely many steps. In Mathematics, especially in the fields of Group theory and Lie theory, a central series is a kind of Normal series of Subgroups or

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. Equivalently, the nilpotency class of G equals the length of the lower central series or upper central series (the minimum n such that the nth term is the trivial subgroup, respectively whole group). If a group has nilpotency class at most m, then it is sometimes called a nil-m group.

The trivial group is the unique group of nilpotency class 0, and groups of nilpotency class 1 are exactly non-trivial abelian groups.

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 $\operatorname{ad}_g(x) := [g,x]$ 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 $G$ . In Mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its In Mathematics, an element x of a ring R is called nilpotent if there exists some positive Integer n such that

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,^[1] and need not be nilpotent in general. In Mathematics, an element x of a Lie group or a Lie algebra is called an n -Engel element, named after Friedrich Engel, They are proven to be nilpotent if they have finite order, and are conjectured to be nilpotent as long as they are finitely generated. In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the

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

Examples

As noted above, every abelian group is nilpotent.
Finite p-groups are nilpotent (proof). In Mathematics, given a Prime number p, a p -group is a Periodic group in which each element has a power of p In Mathematics, given a Prime number p, a p -group is a Periodic group in which each element has a power of p
For a small non-abelian example, consider the quaternion group Q₈, which is a smallest non-abelian p-group. In Group theory, the quaternion group is a non-abelian group of order 8 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.
The Heisenberg group is another example of non-abelian nilpotent group. In Mathematics, the term Heisenberg group, named after Werner Heisenberg, refers to the group of 3×3 upper triangular matrices of the

Properties

Since each successive factor group Z_i+1/Z_i is abelian, and the series is finite, every nilpotent group is a solvable group with a relatively simple structure. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally

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. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function

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 N_G(H) (the normalizer of H in G). In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Group theory, the centralizer and normalizer of a Subset S of a group G are Subgroups of G which 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. In Mathematics, one can often define a direct product of objectsalready known giving a new one In Mathematics, specifically Group theory, the Sylow theorems, named after Ludwig Sylow, form a partial converse to Lagrange's theorem, which

The last statement can be extended to infinite groups: If G is a nilpotent group, then every Sylow subgroup G_p of G is normal, and the direct sum of these Sylow subgroups is the subgroup of all elements of finite order in G (see torsion subgroup). In the theory of Abelian groups the torsion subgroup AT of an abelian group A is the Subgroup of A consisting of all elements

Many properties of nilpotent groups are shared by hypercentral groups. In Mathematics, especially in the fields of Group theory and Lie theory, a central series is a kind of Normal series of Subgroups or

References

^ For the term, compare Engel's theorem, also on nilpotency. In Representation theory, Engel's theorem is one of the basic theorems in the theory of Lie algebras it asserts that for a Lie algebra two concepts of nilpotency

Homology in group theory, by Urs Stammbach, Lecture Notes in Mathematics, Volume 359, Springer-Verlag, New York, 1973, vii+183 pp. review

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Contents

Definition

Explanation of term

Examples

Properties

References