Nilpotency class

This article defines an arithmetic function on a restricted class of groups, namely: nilpotent groups

Definition

Symbol-free definition

For a nilpotent group, the nilpotency class or nilpotence class is defined in any of the following equivalent ways:

• It is the length of the upper central series.
• It is the length of the lower central series.
• It is the minimum possible length of a central series.

A group is said to be of class $c$ if its nilpotency class is less than or equal to $c$.

Equivalence of definitions

For full proof, refer: Equivalence of definitions of nilpotency class

Facts

Relation with solvable length

Further information: Nilpotency class versus derived length

Any nilpotent group is solvable, and there are numerical relations between the nilpotency class and derived length:

• Derived length is logarithmically bounded by nilpotency class
• Derived length gives no upper bound on nilpotency class: For a derived length greater than $1$, the value of the solvable length gives no upper bound on the value of the nilpotency class.