# Second half of lower central series of nilpotent group comprises abelian groups

From Groupprops

## Statement

Suppose is a nilpotent group of nilpotency class . Define the lower central series of as follows:

.

Then, for , is an abelian group, In particular, is an abelian characteristic subgroup.

## Facts used

- Lower central series is strongly central: This states that .

### Applications

- Penultimate term of lower central series is abelian in nilpotent group of class at least three
- Derived length is logarithmically bounded by nilpotency class
- Nilpotent and every abelian characteristic subgroup is central implies class at most two

### Breakdown for upper central series

The first half of the upper central series of a nilpotent group need not comprise Abelian groups. In fact, even the second term of the series need not be Abelian, however large the nilpotence class. More specifically:

- Upper central series may be tight with respect to nilpotence class: For any natural number , we can construct a nilpotent group such that the term of the upper central series of the group has nilpotence class precisely (note: the nilpotence class clearly cannot be
*greater*than , and this result says that tightness may hold. - Second term of upper central series not is Abelian

## Proof

**Given**: A group of nilpotency class .

**To prove**: is Abelian for .

**Proof**: By fact (1), , and since has class , is trivial. Thus, is trivial, and thus, is an abelian group.