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 nilpotence 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 nilpotence class and solvable length:

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