Nilpotent group

In 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. 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.

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

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

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