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Upper central series

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This article defines a quotient-iterated series with respect to the following subgroup-defining function: center

Contents

Definition

Definition with symbols

The upper central series of a group G is an ascending chain of subgroups indexed by ordinals, where the αth member is denoted as Zα(G). It is defined as follows:

  • Z0(G) is the trivial subgroup.
  • When α = β + 1 is a successor ordinal, Zα(G) is the inverse image of the center of G / Zβ(G) with respect to the natural projection map G \to G/Z^{\beta}(G). In other words, \! Z^\alpha(G)/Z^\beta(G) = Z(G/Z^\beta(G)).
  • When α is a limit ordinal, Zα(G) is the union of all Zγ(G) where γ < α.

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

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

Z^0(G) = \{ e \}, \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

Property theory

Member-wise property theory

Each ordinal α gives a subgroup-defining function that sends G to Zα(G). Since each member of the upper central series is defined using a subgroup-defining function, it is characteristic. In fact, from the way we have defined the members, each member is also strictly characteristic.

For full proof, refer: Upper central series members are strictly characteristic

Related subgroup-defining functions

Subgroup-defining function Role in upper central series
center first member, i.e., Z1
second center second member, i.e., Z2
hypercenter the final stable member, i.e., the Zα(G) such that Zα + 1(G) = Zα(G) (α may be transfinite; however, it must exist).

Related group properties

property meaning in terms of upper central series
Centerless group upper central series stays stuck at trivial subgroup; never gets off the ground
Nilpotent group upper central series reaches whole group in finitely many steps
Hypercentral group transfinite upper central series reaches whole group

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.

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