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 is... |
---|---|---|---|
1 | upper central series | the length of the upper central series | the smallest nonnegative integer such that . Here, we define inductively as follows:
is the inverse image of the center under the natural quotient map from to , and is the trivial subgroup. |
2 | lower central series | the length of the lower central series | the smallest nonnegative integer such that is trivial where is repeated times. Here, denotes the commutator of two subgroups. In other words, the lower central series of reaches the identity in finitely many steps. |
3 | central series | the minimum possible length of a central series | the smallest nonnegative integer and a chain of subgroups: such that each is a normal subgroup of and is in the center of . In other words, there exists a central series for of length . |
4 | diagonal subnormal in square | the subnormal depth of the diagonal subgroup in the square of the group | the subnormal depth of the subgroup in the square with subnormal depth . |
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 such that any commutator of the form takes value the identity element, where the are (possibly repeated) elements of . |
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 such that any iterated commutator that involves at least commutator operations (so original inputs) takes value the identity element. [SHOW MORE] |
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 for ) the smallest finite length such that any iterated commutator that involves at least commutator operations (so original inputs) and where the inputs are from , always takes value the identity element. [SHOW MORE] |
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.