Finite p-group of characteristic rank one
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
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.