Groupprops, The Group Properties Wiki (pre-alpha)

From Groupprops

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

Contents 1 Definition 1.1 Symbol-free definition 1.2 Equivalence of definitions 2 Example 3 Facts 3.1 Relation with derived length

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

Example

Any group of prime power order is nilpotent. Further information: prime power order implies nilpotent

A group of order pⁿ, with p prime, can have any nilpotency class between 1 and n − 1 if . For more information of the number of p-groups of various nilpotency class values for various primes, refer nilpotency class distribution of p-groups.

Facts

Relation with derived 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.