Nilpotency class
This article defines an arithmetic function on a restricted class of groups, namely: nilpotent groups
Contents
Definition
Equivalent definitions in tabular format
The notion of nilpotency class (also called nilpotence class) makes sense for any nilpotent group and is a nonnegative integer dependent on the group.
No. | Shorthand | The nilpotency class is defined as ... | The nilpotency class of a nilpotent group ![]() |
---|---|---|---|
1 | upper central series | the length of the upper central series | the smallest nonnegative integer ![]() ![]() ![]()
|
2 | lower central series | the length of the lower central series | the smallest nonnegative integer ![]() ![]() ![]() ![]() ![]() ![]() |
3 | central series | the minimum possible length of a central series | the smallest nonnegative integer ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | diagonal subnormal in square | the subnormal depth of the diagonal subgroup in the square of the group | the subnormal depth of the subgroup ![]() ![]() ![]() |
5 | iterated left-normed commutators trivial | the smallest length such that any iterated left-normed commutator of length more than that becomes trivial | the smallest length ![]() ![]() ![]() ![]() |
6 | iterated commutators of any form trivial | the smallest length such that any iterated commutator (with any kind of parenthesization of terms) of length more than that becomes trivial | the smallest length ![]() ![]() ![]() |
7 | iterated left-normed commutators trivial (generating set version) | (pick a generating set for the group) the smallest finite length such that any iterated commutator (with any kind of parenthesization of terms) of length more than that becomes trivial | (pick a generating set ![]() ![]() ![]() ![]() ![]() ![]() |
Equivalence of definitions
For full proof, refer: Equivalence of definitions of nilpotency class
Terminology
When we say that a group has nilpotency class , we usually mean that the nilpotency class of the group is at most equal to
. If we want to say that the class is exactly
, this is stated explicitly.
Particular cases
Value of ![]() |
Name for groups of nilpotency class at most ![]() |
---|---|
0 | trivial group only |
1 | abelian groups (note that nontrivial abelian groups have class exactly 1) |
2 | group of nilpotency class two. The non-abelian ones among these have class exactly 2 |
3 | group of nilpotency class three |
Example
Any group of prime power order is nilpotent. Further information: prime power order implies nilpotent
A group of order , with
prime, can have any nilpotency class between 1 and
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.
Relation with other arithmetic functions
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
, the value of the derived length gives no upper bound on the value of the nilpotency class.