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:

• $C_{G}(H)\leq H$
• $Z(H)=C_{G}(H)$

Note that the term self-centralizing subgroup is often used for an abelian self-centralizing subgroup, i.e., a subgroup that equals, rather than merely contains, its centralizer. This is equivalent to being maximal among abelian subgroups.

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.

Testing

GAP code

One can write code to test this subgroup property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
View the GAP code for testing this subgroup property at: IsSelfCentralizing
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GAP-codable subgroup property

A short piece of GAP code can test whether a subgroup of a group is self-centralizing: the code is available at GAP:IsSelfCentralizing.