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, 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,, where denotes the commutator subgroup generated by all elements of the form, with g in G and h in H. Since, the subgroup is normal in G for each i. Thus, we can rephrase the 'central' condition above as: is normal in G and is central in for each i. As a consequence, is abelian for each i.
A central series is analogous in Lie theory to a flag that is strictly preserved by the adjoint action ; 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 A0 =, the center Z satisfies A1 ≤ Z. Therefore, the maximal choice for A1 is A1 = Z. Continuing in this way to choose the largest possible Ai + 1 given Ai produces what is called the upper central series. Dually, since An = G, the commutator subgroup satisfies = ≤ An − 1. Therefore, the minimal choice for An − 1 is . Continuing to choose Ai minimally given Ai + 1 such that ≤ 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, these procedures will yield central series.

Lower central series

The lower central series of a group G is the descending series of subgroups
where each Gn + 1 = , the subgroup of G generated by all commutators with x in Gn and y in G. Thus, G2 = = G, the derived subgroup of G; G3 = , G], etc. The lower central series is often denoted γn = Gn.
This should not be confused with the derived series, whose terms are G := , not Gn := . The series are related by G ≤ Gn. For instance, the symmetric group S3 is solvable of class 2: the derived series is S3 ⊵ ⊵. But it is not nilpotent: its lower central series S3 ⊵ ⊵ ⊵ ⋯ 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 Gλ = ∩. If Gλ = 1 for some ordinal λ, then G is said to be a hypocentral group. For every ordinal λ, there is a group G such that Gλ = 1, but Gα ≠ 1 for all α < λ,.
If ω is the first infinite ordinal, then Gω 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 groups 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 Gω = Gn for some finite n, then Gω is the smallest normal subgroup of G with nilpotent quotient, and Gω is called the nilpotent residual of G. This is always the case for a finite group, and defines the F1 term in the lower Fitting series for G.
If Gω ≠ Gn for all finite n, then G/Gω 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.

Upper central series

The upper central series of a group G is the sequence of subgroups
where each successive group is defined by:
and is called the ith center of G. In this case, Z1 is the center of G, and for each successive group, the factor group Zi + 1/Zi is the center of G/Zi, 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 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, elements of coprime order commute, and periodic hypercentral groups are the direct sum of their Sylow p-subgroups. For every ordinal λ there is a group G with Zλ = G, but Zα ≠ G for α < λ, and.

Connection between lower and upper central series

There are various connections between the lower central series and upper central series , particularly for nilpotent groups.
Most simply, a group is abelian if and only if the LCS terminates at the first step if and only if the UCS stabilizes at the first step. More generally, for a nilpotent group, the length of the LCS and the length of the UCS agree. 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, the UCS and LCS of their direct product C2 × Q8 do not: its lower central series is C2 × Q8 ⊵ × ⊵ ×, while the upper central series is C2 × Q8 ⊵ C2 × ⊵ ×.
However, the LCS stabilizes at the zeroth step if and only if it is perfect, while the UCS stabilizes at the zeroth step if and only if it is centerless, which are distinct concepts, and show that the lengths of the LCS and UCS 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-groups, 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 groups, or what is the same, have exponent p. There is a unique most quickly descending such series, the lower exponent-p central series λ defined by:
The second term, λ2, is equal to Gp = Φ, 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 denotes the subgroup generated by the set of central elements of H of order dividing p. The first term, S1, is the subgroup generated by the minimal normal subgroups and so is equal to the 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 S. A. Jennings who used the series to describe the Loewy series of the modular group ring of a p-group.