Upper central series may be tight with respect to nilpotency class
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
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
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
.