Central series

In

matrices with constant diagonal.

This article uses the language of group theory; analogous terms are used for Lie algebras.

A general group possesses a lower central series and upper central series (also called the descending central series and ascending central series, respectively), but these are central series in the strict sense (terminating in the trivial subgroup) if and only if the group is

.

Definition

A central series is a sequence of subgroups

${\displaystyle \{1\}=A_{0}\triangleleft A_{1}\triangleleft \dots \triangleleft A_{n}=G}$

such that the successive quotients are

central

; that is, ${\displaystyle [G,A_{i+1}]\leq A_{i}}$, where ${\displaystyle [G,H]}$ denotes the commutator subgroup generated by all elements of the form ${\displaystyle [g,h]=g^{-1}h^{-1}gh}$, with g in G and h in H. Since ${\displaystyle [G,A_{i+1}]\leq A_{i}\leq A_{i+1}}$, the subgroup ${\displaystyle A_{i+1}}$ is normal in G for each i. Thus, we can rephrase the 'central' condition above as: ${\displaystyle A_{i}}$ is normal in G and ${\displaystyle A_{i+1}/A_{i}}$ is central in ${\displaystyle G/A_{i}}$ for each i. As a consequence, ${\displaystyle A_{i+1}/A_{i}}$ is abelian for each i.

A central series is analogous in

.

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 = {1}, the center Z(G) satisfies A₁ ≤ Z(G). Therefore, the maximal choice for A₁ is 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, for each n,

${\displaystyle G_{n+1}=[G_{n},G]}$,

the subgroup of G generated by all commutators ${\displaystyle [x,y]}$ with ${\displaystyle x\in G_{n}}$ and ${\displaystyle y\in G}$. Thus, ${\displaystyle G_{2}=[G,G]=G^{(1)}}$, the

of G, while ${\displaystyle G_{3}=[[G,G],G]}$, etc. The lower central series is often denoted ${\displaystyle \gamma _{n}(G)=G_{n}}$. We say the series terminates or stablizes when ${\displaystyle G_{n}=G_{n+1}=G_{n+2}=\cdots }$, and the smallest such n is the length of the series.

This should not be confused with the

, whose terms are

${\displaystyle G^{(n)}:=[G^{(n-1)},G^{(n-1)}]}$,

not ${\displaystyle G_{n}=[G_{n-1},G]}$. The two series are related by ${\displaystyle G^{(n)}\leq G_{n}}$. For instance, the symmetric group S₃ is 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)} does not terminate in {e}. 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

λ, define

${\displaystyle G_{\lambda }=\bigcap \{G_{\alpha }:\alpha <\lambda \}}$.

If ${\displaystyle G_{\lambda }=1}$ for some ordinal λ, then G is said to be a hypocentral group. For every ordinal λ, there is a group G such that ${\displaystyle G_{\lambda }=1}$, but ${\displaystyle G_{\alpha }\neq 1}$ for all ${\displaystyle \alpha <\lambda }$, (Malcev 1949).

If ${\displaystyle \omega }$ is the first infinite ordinal, then ${\displaystyle G_{\omega }}$ is the smallest normal subgroup of G such that the quotient is

, Ch. 11).

If ${\displaystyle G_{\omega }=G_{n}}$ for some finite n, then ${\displaystyle G_{\omega }}$ is the smallest normal subgroup of G with nilpotent quotient, and ${\displaystyle G_{\omega }}$ is called the nilpotent residual of G. This is always the case for a finite group, and defines the ${\displaystyle F_{1}(G)}$ term in the

lower Fitting series

for G.

If ${\displaystyle G_{\omega }\neq G_{n}}$ for all finite n, then ${\displaystyle G/G_{\omega }}$ 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

${\displaystyle 1=Z_{0}\triangleleft Z_{1}\triangleleft \cdots \triangleleft Z_{i}\triangleleft \cdots ,}$

where each successive group is defined by:

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

and is called the ith center of G (respectively, second center, third center, etc.). In this case, ${\displaystyle Z_{1}}$ is the

${\displaystyle Z_{i+1}/Z_{i}}$ is the center of ${\displaystyle G/Z_{i}}$, and is called an upper central series quotient. Again, we say the series terminates if it stabilizes into a chain of equalities, and its length is the number of distinct groups in it.

For infinite groups, one can continue the upper central series to infinite

λ, define

${\displaystyle Z_{\lambda }(G)=\bigcup _{\alpha <\lambda }Z_{\alpha }(G).}$

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

).

Connection between lower and upper central series

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

For a nilpotent group, the lengths of the LCS and the UCS agree, and this length 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 ⊵ {e} and quaternion group Q₈ ⊵ {1, −1} ⊵ {1}, the UCS and LCS of their direct product C₂ × Q₈ do not agree: its LCS is C₂ × Q₈ ⊵ {e} × {−1, 1} ⊵ {e} × {1}, while its UCS is C₂ × Q₈ ⊵ C₂ × {−1, 1} ⊵ {e} × {1}.

A group is abelian if and only if the LCS terminates at the first step (the commutator subgroup is the entire group), if and only if the UCS terminates at the first step (the center is the entire group).

By contrast, the LCS terminates at the zeroth step if and only if the group is

on two or more generators is centerless, but its LCS does not stabilize until the first infinite ordinal. This shows that the lengths of the LCS and UCS need not agree in general.

Refined central series

In the study of

exponent

p. There is a unique most quickly descending such series, the lower exponent-p central series λ defined by:

${\displaystyle \lambda _{1}(G)=G}$, and

${\displaystyle \lambda _{n+1}(G)=[G,\lambda _{n}(G)](\lambda _{n}(G))^{p}}$.

The second term, ${\displaystyle \lambda _{2}(G)}$, is equal to ${\displaystyle [G,G]G^{p}=\Phi (G)}$, 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

of a p-group.

See also

• Nilpotent series
  , an analogous concept for solvable groups
• Parent-descendant relations for finite p-groups defined by various kinds of central series
• Unipotent group

References

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Central series" – news · newspapers · books · scholar · JSTOR (January 2007) (Learn how and when to remove this message)

• 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,
  JSTOR 2693793
• Gluškov, V. M. (1952), "On the central series of infinite groups", Mat. Sbornik, New Series, 31: 491–496,
  MR 0052427
• MR 0103215
• MR 0032644
• McLain, D. H. (1956), "Remarks on the upper central series of a group", Proc. Glasgow Math. Assoc., 3: 38–44,
  MR 0084498
• Schenkman, Eugene (1975), Group theory, Robert E. Krieger Publishing,
  MR 0460422
  , especially chapter VI.