Central series

In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, this is an explicit expression that the group is a nilpotent group, and for matrix rings, this is an explicit expression that in some basis the matrix ring consists entirely of upper triangular matrices with constant diagonal.
This article uses the language of group theory; analogous terms are used for Lie algebras.
The lower central series and upper central series (also called the descending central series and ascending central series, respectively), are, despite the "central" in their names, central series if and only if a group is nilpotent.
Definition.
A central series is a sequence of subgroups
such that the successive quotients are central; that is, formula_2, where formula_3 denotes the commutator subgroup generated by all elements of the form formula_4, with "g" in "G" and "h" in "H". Since formula_5, the subgroup formula_6 is normal in "G" for each "i". Thus, we can rephrase the 'central' condition above as: formula_7 is normal in "G" and formula_8 is central in formula_9 for each "i". As a consequence, formula_8 is abelian for each "i".
A central series is analogous in Lie theory to a flag that is strictly preserved by the adjoint action (more prosaically, a basis in which each element is represented by a strictly upper triangular matrix); compare Engel's theorem.
A group need not have a central series. In fact, a group has a central series if and only if it is a nilpotent group. If a group has a central series, then there are two central series whose terms are extremal in certain senses. Since "A"0 = {1}, the center "Z"("G") satisfies "A"1 ≤ "Z"("G"). Therefore, the maximal choice for "A"1 is "A"1 = "Z"("G"). Continuing in this way to choose the largest possible "A""i" + 1 given "Ai" produces what is called the upper central series. Dually, since "An" = "G", the commutator subgroup ["G", "G"] satisfies ["G", "G"] = ["G", "An"] ≤ "A""n" − 1. Therefore, the minimal choice for "A""n" − 1 is ["G", "G"]. Continuing to choose "Ai" minimally given "A""i" + 1 such that ["G", "A""i" + 1] ≤ "Ai" produces what is called the lower central series. These series can be constructed for any group, and if a group has a central series (is a nilpotent group), these procedures will yield central series.
Lower central series.
The lower central series (or descending central series) of a group "G" is the descending series of subgroups
where, for each "n",
the subgroup of "G" generated by all commutators formula_12 with formula_13 and formula_14. Thus, formula_15, the derived subgroup of "G", while formula_16, etc. The lower central series is often denoted formula_17.
This should not be confused with the [[derived series]], whose terms are 
not formula_19. The two series are related by formula_20. For instance, the [[symmetric group]] "S3" is [[Solvable group|solvable]] of class 2: the derived series is "S3" ⊵ {"e", (1 2 3), (1 3 2)} ⊵ {"e"}. But it is not nilpotent: its lower central series "S3" ⊵ {"e", (1 2 3), (1 3 2)} ⊵ {"e", (1 2 3), (1 3 2)} ⊵ ⋯ does not terminate. A nilpotent group is a [[solvable group]], and its derived length is logarithmic in its nilpotency class .
For infinite groups, one can continue the lower central series to infinite [[ordinal numbers]] via [[transfinite recursion]]: for a [[limit ordinal]] "λ", define 
If formula_22 for some ordinal "λ", then "G" is said to be a hypocentral group. For every ordinal "λ", there is a group "G" such that formula_22, but formula_24 for all formula_25, .
If formula_26 is the first infinite ordinal, then formula_27 is the smallest normal subgroup of "G" such that the quotient is [[residually nilpotent]], that is, such that every non-identity element has a non-identity homomorphic image in a nilpotent group . In the field of [[combinatorial group theory]], it is an important and early result that [[free group]]s are residually nilpotent. In fact the quotients of the lower central series are free abelian groups with a natural basis defined by basic commutators, .
If formula_28 for some finite "n", then formula_27 is the smallest normal subgroup of "G" with nilpotent quotient, and formula_27 is called the nilpotent residual of "G". This is always the case for a finite group, and defines the formula_31 term in the [[lower Fitting series]] for "G".
If formula_32 for all finite "n", then formula_33 is not nilpotent, but it is [[residually nilpotent]].
There is no general term for the intersection of all terms of the transfinite lower central series, analogous to the hypercenter (below).
Upper central series.
The upper central series (or ascending central series) of a group "G" is the sequence of subgroups
where each successive group is defined by:
and is called the [[Center (group theory)#Higher centers|"i"th center]] of "G" (respectively, second center, third center, etc.). In this case, formula_36 is the [[center (group theory)|center]] of "G", and for each successive group, the [[factor group]] formula_37 is the center of formula_38, and is called an upper central series quotient.
For infinite groups, one can continue the upper central series to infinite [[ordinal numbers]] via [[transfinite recursion]]: for a [[limit ordinal]] "λ", define 
The limit of this process (the union of the higher centers) is called the hypercenter of the group.
If the transfinite upper central series stabilizes at the whole group, then the group is called hypercentral. Hypercentral groups enjoy many properties of nilpotent groups, such as the normalizer condition (the normalizer of a proper subgroup properly contains the subgroup), elements of coprime order commute, and [[periodic group|periodic]] hypercentral groups are the [[direct sum of groups|direct sum]] of their [[Sylow subgroup|Sylow "p"-subgroups]] . For every ordinal "λ" there is a group "G" with "Z""λ"("G") = "G", but "Z""α"("G") ≠ "G" for "α" < "λ", and .
Connection between lower and upper central series.
There are various connections between the lower central series (LCS) and upper central series (UCS) , particularly for [[nilpotent group]]s.
Most simply, a group is abelian if and only if the LCS terminates at the first step (the commutator subgroup is trivial) if and only if the UCS stabilizes at the first step (the center is the entire group). More generally, for a nilpotent group, the length of the LCS and the length of the UCS agree (and is called the nilpotency class of the group). However, the LCS and UCS of a nilpotent group may not necessarily have the same terms. For example, while the UCS and LCS agree for the [[cyclic group]] "C2" and [[quaternion group]] "Q8" (which are "C2" ⊵ {"e"} and "Q8" ⊵ {1, -1} ⊵ {1} respectively), the UCS and LCS of their [[direct product]] "C2" × "Q8" do not: its lower central series is "C2" × "Q8" ⊵ {"e"} × {-1, 1} ⊵ {"e"} × {1}, while the upper central series is "C2" × "Q8" ⊵ "C2" × {-1, 1} ⊵ {"e"} × {1}.
However, the LCS stabilizes at the zeroth step if and only if it is [[perfect group|perfect]], while the UCS stabilizes at the zeroth step if and only if it is [[centerless group|centerless]], which are distinct concepts, and show that the lengths of the LCS and UCS (interpreted to mean the length before stabilization) need not agree in general.
For a perfect group, the UCS always stabilizes by the first step, a fact called [[Grün's lemma]]. However, a centerless group may have a very long lower central series: a [[free group]] on two or more generators is centerless, but its lower central series does not stabilize until the first infinite ordinal.
Refined central series.
In the study of [[p-group|"p"-group]]s, it is often important to use longer central series. An important class of such central series are the exponent-"p" central series; that is, a central series whose quotients are [[elementary abelian group]]s, or what is the same, have [[exponent (group theory)|exponent]] "p". There is a unique most quickly descending such series, the lower exponent-"p" central series λ defined by:
The second term, formula_42, is equal to formula_43, the [[Frattini subgroup]]. The lower exponent-"p" central series is sometimes simply called the "p"-central series.
There is a unique most quickly ascending such series, the upper exponent-"p" central series S defined by:
where Ω("Z"("H")) denotes the subgroup generated by (and equal to) the set of central elements of "H" of order dividing "p". The first term, S1("G"), is the subgroup generated by the minimal normal subgroups and so is equal to the [[socle (mathematics)|socle]] of "G". For this reason the upper exponent-"p" central series is sometimes known as the socle series or even the Loewy series, though the latter is usually used to indicate a descending series.
Sometimes other refinements of the central series are useful, such as the Jennings series "κ" defined by:
The Jennings series is named after [[Stephen Arthur Jennings]] who used the series to describe the [[Loewy series]] of the modular [[group ring]] of a "p"-group.
References.
[[Category:Subgroup series]]