Lower exponent-p central series

Definition
Suppose $$p$$ is a prime number and $$G$$ is a finite p-group. The lower exponent-p central series, also called the p-central series, of $$G$$ is a series $$\lambda_n(G)$$, $$n \in \mathbb{N}$$, defined as follows:


 * $$\lambda_1(G) = G$$
 * $$\lambda_{n+1}(G) = [G,\lambda_n(G)]\mho^1(\lambda_n(G))$$

Here, $$\mho^1(\lambda_n(G)) = (\lambda_n(G))^p$$ is the subgroup generated by the $$p^{th}$$ powers of the elements from $$\lambda_n(G)$$.

It is the fastest descending exponent-p central series.

Corresponding ascending series
For a finite p-group, the corresponding ascending series, the upper exponent-p central series, is the socle series.

Other related series
The following series are closely related:


 * Frattini series
 * Jennings series