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}.

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.

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.

The definition uses the idea of a central series for a group.

The following are equivalent definitions for a nilpotent group G: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

Equivalently, the nilpotency class of G equals the length of the lower central series or upper central series.

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.

[2][3] The natural numbers k for which any group of order k is nilpotent have been characterized (sequence A056867 in the OEIS).

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).

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

The following statements are equivalent for finite groups,[10] revealing some useful properties of nilpotency:Proof: Statement (d) 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).