Exponent-p central series

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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:

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.