Central Series - Refined Central Series

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) = (λ_n(G))p

The second term, λ₂(G), is equal to Gp = 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) = (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.