Nilpotency class: Difference between revisions
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For a [[nilpotent group]], the '''nilpotency class''' or '''nilpotence class''' is defined in any of the following equivalent ways: | 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 [[defining ingredient::upper central series]]. | ||
* It is the length of the [[lower central series]] | * It is the length of the [[defining ingredient::lower central series]]. | ||
* It is the minimum possible length of a central series | * It is the minimum possible length of a [[defining ingredient::central series]]. | ||
A group is said to be of class <math>c</math> if its | A group is said to be of class <math>c</math> if its nilpotency class is less than or equal to <math>c</math>. | ||
===Equivalence of definitions=== | ===Equivalence of definitions=== | ||
Revision as of 10:42, 29 December 2009
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 if its nilpotency class is less than or equal to .
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 , the value of the solvable length gives no upper bound on the value of the nilpotence class.