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} =, the subgroup of G generated by all commutators with x in G_n and y in G. Thus, G₂ = = G(1), the derived subgroup of G; G₃ =, 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) :=, not G_n := . 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).