C-closed self-centralizing subgroup

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: c-closed subgroup and self-centralizing subgroup
View other subgroup property conjunctions | view all subgroup properties

Definition

Symbol-free definition

A subgroup of a group is termed a c-closed self-centralizing subgroup or a centralizer of Abelian subgroup if it satisfies the following equivalent conditions:

1. Its centralizer equals its center and it equals the centralizer of its center (all relative to the whole group).
2. It is self-centralizing (i.e., it contains its own centralizer in the whole group) and also occurs as the centralizer of some subgroup of the whole group.
3. It occurs as the centralizer of some Abelian subgroup of the whole group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed a c-closed self-centralizing subgroup or a centralizer of Abelian subgroup if it satisfies the following equivalent conditions:

1. $C_{G}(H)=Z(H)$ and $C_{G}(Z(H))=H$.
2. $C_{G}(H)\leq H$ and there exists a subgroup $K\leq G$ such that $H=C_{G}(K)$.
3. There exists an Abelian subgroup $K$ of $G$ such that $H=C_{G}(K)$.

Relation with other properties

Stronger properties

• c-closed critical subgroup
• Maximal among abelian subgroups

Weaker properties

• c-closed subgroup
• Self-centralizing subgroup