Upper central series

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

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:

Case for ordinal $\alpha$	Verbal definition of $\alpha^{th}$ member of upper central series	Definition using the $Z^\alpha(G)$ notation
$\alpha = 0$	trivial subgroup	$Z^0(G) = 1$
$\alpha = \beta + 1$ is a successor ordinal. Note that this includes all positive integer values of $\alpha$.	its image modulo the previous member is the center of the quotient group by the previous member.	$Z^{\alpha}(G)$ is the inverse image of the center of $G/Z^{\beta}(G)$ with respect to the natural projection map $G \to G/Z^{\beta}(G)$. In other words, $\! Z^\alpha(G)/Z^\beta(G) = Z(G/Z^\beta(G))$.
$\alpha$ is a limit ordinal.	union of all the previous members	$Z^\alpha(G) = \bigcup_{\beta < \alpha} Z^\beta(G)$. Equivalently, $Z^\alpha(G) = \langle Z^\beta(G) \rangle_{\beta < \alpha}$

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

Subgroup properties

Subgroup property	Meaning	Satisfied?	Key facts/examples	Restricting to groups with certain properties	Relevant proof method
characteristic subgroup	invariant under all automorphisms	Yes	subgroup-defining function value is characteristic	--	--
normal subgroup	invariant under all inner automorphisms	Yes	follows from being characteristic and characteristic implies normal	--	--
strictly characteristic subgroup	invariant under all surjective endomorphisms	Yes	Combine center is strictly characteristic with strict characteristicity is quotient-transitive	--	inductive proof methods for the ascending series corresponding to a subgroup-defining function#Using quotient-transitivity for subgroup properties
quotient-powering-invariant subgroup	the subgroup is normal, and if the whole group is powered over a set of primes, so is the quotient.	Yes	Combine center is quotient-powering-invariant with quotient-powering-invariance is quotient-transitive	--	inductive proof methods for the ascending series corresponding to a subgroup-defining function#Using quotient-transitivity for subgroup properties
local powering-invariant subgroup	if an element of the subgroup has a unique $n^{th}$ root in the group, that root is in the subgroup.	No	See second center not is local powering-invariant	true for nilpotent groups, see upper central series members are local powering-invariant in nilpotent group. Follows for these from center is local powering-invariant and local powering-invariance is quotient-transitive in nilpotent group	inductive proof methods for the ascending series corresponding to a subgroup-defining function#Using quotient-transitivity for subgroup properties
completely divisibility-closed subgroup	if the whole group is $p$-divisible for some prime $p$, all $p^{th}$ roots of elements in the subgroup lie in the subgroup.	No	Follows from center not is divisibility-closed	true for nilpotent groups, see upper central series members are completely divisibility-closed in nilpotent group.	downward induction on upper central series
fully invariant subgroup	invariant under all endomorphisms	No	Follows from center not is fully invariant	No meaningful class of examples. See center not is fully invariant in odd-order class two p-group	--

Related subgroup-defining functions

Subgroup-defining function	Role in upper central series
center	first member, i.e., $Z^1$
second center	second member, i.e., $Z^2$
hypercenter	the final stable member, i.e., the $Z^\alpha(G)$ such that $Z^{\alpha + 1}(G) = Z^\alpha(G)$ ($\alpha$ 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)