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

Contents 1 Definition 1.1 Definition with symbols 2 Property theory 2.1 Member-wise property theory 2.2 Related subgroup-defining functions 2.3 Related group properties 2.4 Relation with lower central series

Definition

Definition with symbols

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

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

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

Property theory

Member-wise property theory

Each ordinal α gives a subgroup-defining function that sends G to Z^α(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, and nilpotent not implies UL-equivalent.

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

Nature of fact	Fact for lower central series	Fact for upper central series
Is the series a strongly central series?	Lower central series is strongly central	upper central series not is strongly central (i.e., the upper central series need not always be a strongly central series).
What is the nilpotency class of the members of the series?	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	Upper central series may be tight with respect to nilpotency class
Are the members verbal subgroups and/or fully invariant subgroups in the whole group?	Lower central series members are verbal (and since verbal implies fully invariant, they are also fully invariant)	Upper central series members need not be fully invariant (even for a nilpotent group)