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, the existence of a central series means it is a nilpotent group; for matrix rings (considered as Lie algebras), it means that in some basis the 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.
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 nilpotent. A related but distinct construction is the derived series, which terminates in the trivial subgroup whenever the group is solvable.
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 (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 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 Ai + 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] ≤ An − 1. Therefore, the minimal choice for An − 1 is [G, G]. Continuing to choose Ai minimally given Ai + 1 such that [G, Ai + 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 G = G1 ⊵ G2 ⊵ ⋯ ⊵ Gn ⊵ ⋯, where, for each n, , the subgroup of G generated by all commutators with and . Thus, , the derived subgroup of G, while , etc. The lower central series is often denoted . We say the series terminates or stablizes when , and the smallest such n is the length of the series.
This should not be confused with the derived series, whose terms are
, not . The two series are related by . 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 .
For infinite groups, one can continue the lower central series to infinite ordinal numbers via transfinite recursion: for a limit ordinal λ, define . If for some ordinal λ, then G is said to be a hypocentral group. For every ordinal λ, there is a group G such that , but for all , .
If is the first infinite ordinal, then 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 for some finite n, then is the smallest normal subgroup of G with nilpotent quotient, and is called the nilpotent residual of G. This is always the case for a finite group, and defines the term in the lower Fitting series for G.
If for all finite n, then 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