# 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 nilpotence 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

`Further information: Faithful semidirect product of cyclic p-groups`

Let be an odd prime. Let be the cyclic group of order , and be the cyclic group of order . Consider the action of on , where the generator acts via multiplication by . Let be the semidirect product.

The group in this case is isomorphic to the semidirect product of the cyclic group of order with the cyclic group of order , with the generator acting via multiplication by . This tells us that the upper central series of is precisely the first terms of the upper central series of , as required.