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

Stronger properties

 * Weaker than::Abelian critical subgroup

Weaker properties

 * Stronger than::critical subgroup
 * Stronger than::c-closed self-centralizing subgroup
 * Stronger than::c-closed subgroup