Abelian critical subgroup

From Groupprops
Jump to: navigation, search
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 Abelian critical subgroup