Normal closure-closed subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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Definition

Symbol-free definition

A subgroup property is said to be normal closure-closed if whenever a subgroup has the property in the whole group, its normal closure also has the property.

Definition with symbols

A subgroup property $p$ is said to be normal core-closed if whenever $H$ satisfies property $p$ in $G$ , the normal core $<H^{G}>$ also satisfies $p$ in $G$ .

Relation with other metaproperties

Stronger metaproperties

Join-closed subgroup property
Finite-join-closed subgroup property when we are guaranteed that there are only finitely many conjugates
Subgroup properties stronger than normality
Identity-true subgroup properties stronger than contranormality

Related metaproperties

Opposite metaproperties

NCT-subgroup property that is not stronger than the property of being the trivial subgroup: For full proof, refer: NCT and not trivial implies not normal closure-closed
NCI-subgroup property that is not stronger than contranormality: For full proof, refer: NCT and not contranormal implies not normal closure-closed

Subgroup properties that are normal closure-closed

Join-closed subgroup properties

One way of establishing that a subgroup property is normal closure-closed is to show that the join of any family of subgroups having the property, also has the property.

Subgroup properties stronger than normality

If a subgroup property is stronger than normality, then the normal closure of any subgroup having the property is itself, and hence whenever a subgroup has the property, so does its normal closure.