Finite p-group of characteristic rank one

Definition

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

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

Finite $p$-groups of characteristic rank one are completely classified.

Relation with other properties

Stronger properties

• Finite p-group of rank one
• Finite p-group of normal rank one

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