Upper central series

Definition
The upper central series of a group $$G$$ is an ascending chain of subgroups indexed by ordinals (including zero), where the $$\alpha^{th}$$ member is denoted as $$Z^{\alpha}(G)$$. It is defined as follows:

The zeroth member is the trivial group, the first member is the center, and the second member is the second center. In other words:

$$Z^0(G) \le Z^1(G) \le Z^2(G) \le \dots$$

$$Z^0(G) = 1, \qquad Z^1(G) = Z(G), \qquad Z^2(G)/Z^1(G) = Z(G/Z^1(G)), \qquad Z^3(G)/Z^2(G) = Z(G/Z^2(G)), \dots$$

$$Z^\omega(G) = \bigcup_{n \in \mathbb{N}} Z^n(G)$$

$$\! Z^{\omega + 1}(G)/Z^\omega(G) = Z(G/Z^\omega(G)), Z^{\omega + 2}(G)/Z^{\omega + 1}(G) = Z(G/Z^{\omega + 1}(G)), \dots$$

Proof methods
The following articles give a sense of the important methods used to prove facts about the upper central series:


 * Upward induction for upper central series
 * Downward induction for upper central series
 * Inductive proof methods for the ascending series corresponding to a subgroup-defining function

Relation with lower central series
For a nilpotent group, the lower central series and upper central series are closely related. They both have the same length, and there is a containment relation between them, which follows from the combination of the facts that upper central series is fastest ascending central series and lower central series is fastest descending central series. However, they need not coincide. Nilpotent groups where they do coincide are termed UL-equivalent groups, and nilpotent not implies UL-equivalent.

Here is a table with some distinctions/contrasts between the two central series: