Upper central series may be tight with respect to nilpotency class
Contents
Statement
Let be any natural number. Then, we can construct a nilpotent group
of nilpotency class
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
,
has nilpotency class precisely
.
Related facts
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:
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
.