Upper central series may be tight with respect to nilpotency class
We can find a with the property that for any , has nilpotency class precisely .
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
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 .