# Finite p-group of characteristic rank one

From Groupprops

## Contents

## Definition

A **finite p-group of characteristic rank one** is defined as a group of prime power order (i.e., a finite -group) satisfying the following equivalent conditions:

- Its characteristic rank is at most one (the characteristic rank can be zero only for the trivial group, otherwise it is one)
- Every Abelian characteristic subgroup of it is cyclic.

Finite -groups of characteristic rank one are completely classified.

## Relation with other properties

### Stronger properties

## Metaproperties

### Characteristic subgroups

This group property is characteristic subgroup-closed: any characteristic subgroup of a group with the property, also has the property

View characteristic subgroup-closed group properties]]

If is a finite -group of characteristic rank one, then every characteristic subgroup of also has characteristic rank one.