# 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.

## Relation with other 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:

## Subgroup series properties

Property Meaning Satisfied? Proof
fully invariant series all the member subgroups are fully invariant subgroups Yes lower exponent-p central series is fully invariant
strongly central series descending series $G_m$ where $[G_m,G_n] \le G_{m+n}$ for all $m,n$ Yes lower exponent-p central series is strongly central