Cocentral subgroup of normalizer

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 a cocentral subgroup of normalizer if it is a cocentral subgroup of its normalizer.

Definition with symbols

A subgroup $H$ of a group $G$ is said to be a cocentral subgroup of normalizer if $HZ(N_{G}(H))=N_{G}(H)$ where $Z(N_{G}(H))$ denotes the center of $N_{G}(H)$ and $N_{G}(H)$ denotes the normalizer of $H$.

Formalisms

In terms of the in-normalizer operator

This property is obtained by applying the in-normalizer operator to the property: cocentral subgroup
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Relation with other properties

Stronger properties

• Cocentral subgroup
• Weakly cocentral subgroup
• Self-normalizing subgroup

Weaker properties

• Central factor of normalizer

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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