Abelian critical subgroup

This article describes a property that arises as the conjunction of a subgroup property: critical subgroup with a group property (itself viewed as a subgroup property): Abelian group
View a complete list of such conjunctions

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 $G$

Equivalence of definitions

For full proof, refer: Equivalence of definitions of critical subgroup