# Exponent-p central series

From Groupprops

## Definition

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

An **exponent-p central series** of is a subgroup series:

satisfying the following conditions:

- It is a normal series: each is normal in .
- Each quotient is contained in the socle of , i.e., is a central subgroup of
**and**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.