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

## Statement

Let be any natural number. Then, we can construct a Nilpotent group (?) 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 , the upper central series of is precisely the first terms of the upper central series of . In particular, we can find a with the property that each has nilpotence class precisely : in other words, it is not a group of class .

## Related facts

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

## 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 .