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Upper central series
From Groupprops
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
. In other words,
.
- 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:
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, 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) |