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:

$G=K_{1}\geq K_{2}\geq \dots \geq K_{n}=1$

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.