C-closed critical subgroup

Definition

A subgroup of a group of prime power order is termed a c-closed critical subgroup if it satisfies the following equivalent conditions:

1. It is a critical subgroup and also a c-closed subgroup: it equals the centralizer of some subgroup of the group.
2. It is a critical subgroup and also a c-closed self-centralizing subgroup: its centralizer equals its center and it equals the centralizer of its center.
3. It is a critical subgroup that occurs as the centralizer of an Abelian characteristic subgroup.

Relation with other properties

Stronger properties

• Abelian critical subgroup

Weaker properties

• critical subgroup
• c-closed self-centralizing subgroup
• c-closed subgroup