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