Abelian critical subgroup

Definition
Let $$G$$ be a group of prime power order. A subgroup $$H$$ of $$G$$ is termed an Abelian critical subgroup if it satisfies the following equivalent conditions:


 * 1) $$H$$ is Abelian as a group, and is a critical subgroup of $$G$$
 * 2) $$H$$ is an Abelian characteristic subgroup of $$G$$ that is also self-centralizing
 * 3) $$H$$ is a maximal among Abelian characteristic subgroups of $$G$$ that is also self-centralizing
 * 4) $$H$$ is a characteristic subgroup that is maximal among Abelian normal subgroups of $$G$$