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 .
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:
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 .