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
λn + 1(G) = (λn(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
Sn+1(G)/Sn(G) = Ω(Z(G/Sn(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
κn + 1(G) = (κi(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.

Read more about this topic:  Central Series

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