Second half of lower central series of nilpotent group comprises abelian groups
From Groupprops
Contents |
Statement
Suppose G is a nilpotent group of nilpotency class c. Define the lower central series of G as follows:
![G_1 = G, \qquad G_{m+1} = [G_m, G]](/w/images/math/4/a/a/4aabc28727c1d0b0b6543f001e060083.png)
.
Then, for 
, Gk is an abelian group, In particular, Gk is an abelian characteristic subgroup.
Facts used
- Lower central series is strongly central: This states that
![[G_m,G_n] \le G_{m+n}](/w/images/math/c/b/4/cb4fc948effac1a4ce8b5ade59e3708f.png)
.
Applications
- Penultimate term of lower central series is abelian in nilpotent group of class at least three
- Solvable length is logarithmically bounded by nilpotence 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 c, we can construct a nilpotent group such that the kth term of the upper central series of the group has nilpotence class precisely k (note: the nilpotence class clearly cannot be greater than k, and this result says that tightness may hold.
- Second term of upper central series not is Abelian
Proof
Given: A group G of nilpotence class c.
To prove: Gk is Abelian for .
Proof: By fact (1), ![[G_k, G_k] \le G_{2k} \ge G_{c+1}](/w/images/math/6/6/0/660bc0681d08c127ad2470bab26c8a55.png)
, and since G has class c, Gc + 1 is trivial. Thus, [Gk,Gk] is trivial, and thus, Gk is an Abelian group.
