Group in which maximal among abelian normal implies self-centralizing

Definition
A group in which maximal among Abelian normal implies self-centralizing is a group with the following property: any subgroup that is maximal among Abelian normal subgroups is a self-centralizing subgroup.

Stronger properties

 * Weaker than::Abelian group
 * Weaker than::Nilpotent group:
 * Weaker than::Supersolvable group:

Weaker properties
If the group is non-Abelian, then it is a group having an Abelian normal non-central subgroup