Normal closure-closed subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
View a complete list of subgroup metaproperties
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VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

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

Related metaproperties

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