Group of prime power order in which every abelian subgroup of maximum order is normal

Definition
Suppose $$G$$ is a group of prime power order. We say that every abelian subgroup of maximum order is normal if every defining ingredient::abelian subgroup of maximum order $$H$$ of $$G$$ is a normal subgroup of $$G$$.

Relation with other properties

 * Group of prime power order having an abelian subgroup of maximum order which is normal