# Upper central series may be tight with respect to nilpotency class

## Contents

## Statement

Let be any natural number. Then, we can construct a nilpotent group of nilpotency class with the following property.

Let denote the member of the Upper central series (?) of : is the center and is the center of for all . By definition of Nilpotency class (?), .

We can find a with the property that for any , has nilpotency class precisely .

## Related facts

### Opposite facts for lower central series

The corresponding statement is not true for the lower central series. Some related facts:

- Lower central series is strongly central
- Second half of lower central series of nilpotent group comprises abelian groups
- Penultimate term of lower central series is abelian in nilpotent group of class at least three

### Opposite facts for upper central series

It is also true that the upper central series for any member of the upper central series (beyond the center) grows faster than the actual upper central series of the whole group. See:

## Proof

Let be groups such that each is a nilpotent group of nilpotency class precisely , i.e., it is *not* nilpotent of class smaller than . Define as the external direct product:

Now, for each , we have:

In particular, we obtain that:

From the given data, in particular the fact that has nilpotency class exactly , it is clear that has nilpotency class exactly .