Normal closure-closed subgroup property

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 $$$$ also satisfies $$p$$ in $$G$$.

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

 * Normalizer-closed subgroup property
 * Normal core-closed subgroup property
 * Characteristic closure-closed subgroup property

Opposite metaproperties

 * NCT-subgroup property that is not stronger than the property of being the trivial subgroup:
 * NCI-subgroup property that is not stronger than contranormality:

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.