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.

Stronger properties

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

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