# Abelian critical subgroup

From Groupprops

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 be a group of prime power order. A subgroup of is termed an **Abelian critical subgroup** if it satisfies the following equivalent conditions:

- is Abelian as a group, and is a critical subgroup of
- is an Abelian characteristic subgroup of that is also self-centralizing
- is a maximal among Abelian characteristic subgroups of that is also self-centralizing
- is a characteristic subgroup that is maximal among Abelian normal subgroups of

### Equivalence of definitions

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