Maximal class
A nilpotent group G is a group that has an upper central series that terminates with G. Provably equivalent definitions include a group that has a central series of finite length or a lower central series that terminates with {1}.
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 of a central series for a group. The following are equivalent definitions for a nilpotent group :
...
Wikipedia