# Lower exponent-p central series

From Groupprops

## Contents

## Definition

Suppose is a prime number and is a finite p-group. The **lower exponent-p central series**, also called the **p-central series**, of is a series , , defined as follows:

Here, is the subgroup generated by the powers of the elements from .

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.

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 where for all | Yes | lower exponent-p central series is strongly central |