Exponent-p central series

Definition
Suppose $$p$$ is a prime number and $$G$$ is a finite p-group, or more generally a nilpotent p-group that has finite exponent.

An exponent-p central series of $$G$$ is a subgroup series:

$$G = K_1 \ge K_2 \ge \dots \ge K_n = 1$$

satisfying the following conditions:


 * It is a normal series: each $$K_i$$ is normal in $$G$$.
 * Each quotient $$K_i/K_{i+1}$$ is contained in the socle of $$G/K_{i+1}$$, i.e., $$K_i/K_{i+1}$$ is a central subgroup of $$G/K_{i+1}$$ and $$K_i/K_{i+1}$$ is an elementary abelian group.

The fastest descending exponent-p central series is termed the lower exponent-p central series. The fastest ascending exponent-p central series is the socle series.