Upper central series may be tight with respect to nilpotency class
Let be any natural number. Then, we can construct a Nilpotent group (?) with the following property.
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 .
The corresponding statement is not true for the lower central series. Some related facts:
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 .