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 characteristic series, which, despite the names, are central series if and only if a group is nilpotent.

Definition

A central series is a sequence of subgroups

\{1\} = A_0 \triangleleft A_1 \triangleleft \dots \triangleleft A_n = G

such that the successive quotients are central; that is, [G, A_{i + 1}] ≤ A_i, where [G, H] denotes the commutator subgroup generated by all g⁻¹h⁻¹gh for g in G and h in H. As [G, A_{i + 1}] ≤ A_i ≤ A_{i + 1}, in particular A_{i + 1} is normal in G for each i, and so equivalently we can rephrase the 'central' condition above as: A_{i + 1}/A_i commutes with all of G/A_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 A₁ ≤ Z(G), the largest choice for A₁ is precisely A₁ = Z(G). Continuing in this way to choose the largest possible A_{i + 1} given A_i produces what is called the upper central series. Dually, since A_n= G, the commutator subgroup [G, G] satisfies [G, G] = [G, A_n] ≤ A_{n − 1}. Therefore the minimal choice for A_{n − 1} is [G, G]. Continuing to choose A_i minimally given A_{i + 1} such that [G, A_{i + 1}] ≤ A_i 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

G = G₁ ⊵ G₂ ⊵ ⋯ ⊵ G_n ⊵ ⋯,

where each G_{n + 1} = [G_n, G], the subgroup of G generated by all commutators [x, y] with x in G_n and y in G. Thus, G₂ = [G, G] = G⁽¹⁾, the derived subgroup of G; G₃ = [[G, G], G], etc. The lower central series is often denoted γ_n(G) = G_n.

This should not be confused with the derived series, whose terms are G⁽ⁿ⁾ := [G⁽ⁿ⁻¹⁾,G⁽ⁿ⁻¹⁾], not G_n := [G_n−1, G]. The series are related by G⁽ⁿ⁾ ≤ G_n. In particular, a nilpotent group is a solvable group, and its derived length is logarithmic in its nilpotency class (Schenkman 1975, p. 201,216).

For infinite groups, one can continue the lower central series to infinite ordinal numbers via transfinite recursion: for a limit ordinal λ, define G_λ = ∩ { G_α : α < λ}. If G_λ = 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 G_λ = 1, but G_α ≠ 1 for all α < λ, (Malcev 1949).

If ω is the first infinite ordinal, then G_ω 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 (Schenkman 1975, p. 175,183). 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, (Hall 1959, Ch. 11).

If G_ω = G_n for some finite n, then G_ω is the smallest normal subgroup of G with nilpotent quotient, and G_ω is called the nilpotent residual of G. This is always the case for a finite group, and defines the F₁(G) term in the lower Fitting series for G.

If G_ω ≠ G_n for all finite n, then G/G_ω 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

1 = Z_0 \triangleleft Z_1 \triangleleft \cdots \triangleleft Z_i \triangleleft \cdots,

where each successive group is defined by:

Z_{i+1} = \{x\in G \mid \forall y\in G:[x,y] \in Z_i \}

and is called the ith center of G (respectively, second center, third center, etc.). In this case, Z₁ is the center of G, and for each successive group, the factor group Z_{i + 1}/Z_i is the center of G/Z_i, 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

Z_\lambda(G) = \bigcup_{\alpha < \lambda} Z_\alpha(G).

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 hypercentral groups are the direct sum of their Sylow p-subgroups (Schenkman 1975, Ch. VI.3). For every ordinal λ there is a group G with Z_λ(G) = G, but Z_α(G) ≠ G for α < λ, (Gluškov 1952) and (McLain 1956).

Connection between lower and upper central series

There are various connections between the lower central series and upper central series (Ellis 2001), particularly for nilpotent groups.

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 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 (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-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:

λ₁(G) = G, and

λ_{n + 1}(G) = [G, λ_n(G)] (λ_n(G))^p

The second term, λ₂(G), is equal to [G, G]G^p = Φ(G), 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:

S₀(G) = 1

S_n+1(G)/S_n(G) = Ω(Z(G/S_n(G)))

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 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:

κ₁(G) = G, and

κ_{n + 1}(G) = [G, κ_n(G)] (κ_i(G))^p, where i is the smallest integer larger than or equal to n/p.

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.

References

Ellis, Graham (October 2001), "On the Relation between Upper Central Quotients and Lower Central Series of a Group", Transactions of the American Mathematical Society, 353 (10): 4219–4234, doi:10.1090/S0002-9947-01-02812-4, JSTOR 2693793
Gluškov, V. M. (1952), "On the central series of infinite groups", Mat. Sbornik N.S., 31: 491–496, MR 0052427
Hall, Marshall (1959), The theory of groups, Macmillan, MR 0103215
Malcev, A. I. (1949), "Generalized nilpotent algebras and their associated groups", Mat. Sbornik N.S., 25 (67): 347–366, MR 0032644
McLain, D. H. (1956), "Remarks on the upper central series of a group", Proc. Glasgow Math. Assoc., 3: 38–44, doi:10.1017/S2040618500033414, MR 0084498
Schenkman, Eugene (1975), Group theory, Robert E. Krieger Publishing, ISBN 978-0-88275-070-5, MR 0460422 , especially chapter VI.

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