Self-centralizing subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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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

Metaproperties

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
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The trivial subgroup and the whole group are clearly self-centralizing.

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.