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, 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.

A central series is a sequence of subgroups

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 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 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.

The lower central series (or descending central series) of a group ${\displaystyle G}$ is the descending series of subgroups