Self-centralizing subgroup

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Definition

Symbol-free definition

A subgroup of a group is said to be self-centralizing if it satisfies the following equivalent conditions:

• It contains its own centralizer in the whole group
• Its center equals its centralizer in the whole group

Definition with symbols

A subgroup $H$ of a group $G$ is said to be self-centralizing if it satisfies the following equivalent conditions:

• $C_{G}(H)\leq H$
• $Z(H)=C_{G}(H)$

Relation with other properties

Stronger properties

• CC-subgroup
• Centralizer-free subgroup
• Self-normalizing subgroup

Under additional conditions:

• In any group, a maximal among Abelian subgroups
• In a supersolvable group or nilpotent group, maximal among Abelian normal subgroups
• In a solvable group, the Fitting subgroup

Weaker properties

• Subgroup containing the center

Examples

For a complete list of examples of self-centralizing subgroups, refer:

Metaproperties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
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If a subgroup is self-centralizing in the whole group, it is also self-centralizing in every intermediate subgroup.

Upward-closedness

This subgroup property is upward-closed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group
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If $H$ is a self-centralizing subgroup of $G$, and $K\leq G$ is a subgroup containing $H$, then $K$ is also a self-centralizing subgroup of $G$.

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
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Since any subgroup containing a self-centralizing subgroup is self-centralizing, a join of any nonempty collection of self-centralizing subgroups is again self-centralizing.