Nilpotent group

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 {1}.

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.

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 Template:Mvar:Template:Unordered list

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

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

File:HeisenbergCayleyGraph.png

A portion of the Cayley graph of the discrete Heisenberg group, a well-known nilpotent group.

• 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.
• The direct product of two nilpotent groups is nilpotent.
• 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.
• 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 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 {1}, Z(G), G.

The natural numbers k for which any group of order k is nilpotent have been characterized (sequence A056867 in the OEIS).

Explanation of term

Nilpotent groups are called so 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 ${\mathrm{ad}}_g:G\to G$ defined by ${\mathrm{ad}}_g(x):=[g,x]$ (where $[g,x]=g^{-1}{x}^{-1}gx$ is the commutator of $g$ and $x$) is nilpotent in the sense that the $n$th iteration of the function is trivial: ${\left({\mathrm{ad}}_g\right)}^n(x)=e$ for all $x$ in $G$.

This is not a defining characteristic of nilpotent groups: groups for which ${\mathrm{ad}}_g$ 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 Z_i+1/Z_i 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 Template:Mvar is nilpotent of class at most Template:Mvar; in addition, if Template:Mvar is a homomorphism of a nilpotent group of class Template:Mvar, then the image of Template:Mvar is nilpotent of class at most Template:Mvar.

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

Many properties of nilpotent groups are shared by hypercentral groups.

References

• review