Second half of lower central series of nilpotent group comprises abelian groups
- Lower central series is strongly central: This states that .
- Penultimate term of lower central series is abelian in nilpotent group of class at least three
- Derived length is logarithmically bounded by nilpotency class
- Nilpotent and every abelian characteristic subgroup is central implies class at most two
Breakdown for upper central series
The first half of the upper central series of a nilpotent group need not comprise Abelian groups. In fact, even the second term of the series need not be Abelian, however large the nilpotence class. More specifically:
- Upper central series may be tight with respect to nilpotence class: For any natural number , we can construct a nilpotent group such that the term of the upper central series of the group has nilpotence class precisely (note: the nilpotence class clearly cannot be greater than , and this result says that tightness may hold.
- Second term of upper central series not is Abelian
Given: A group of nilpotency class .
To prove: is Abelian for .
Proof: By fact (1), , and since has class , is trivial. Thus, is trivial, and thus, is an abelian group.