Characteristic core-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 characteristic core-closed if whenever a subgroup has the proeprty in the whole group, so does its characteristic core (viz the largest characteristic subgroup contained inside it).

Definition with symbols

A subgroup property $p$ is said to be characteristic core-closed if whenever $H$ satisfies property $p$ in $G$, and $K$ is the characteristic core of $H$, $K$ satisfies property $p$ in $G$.

Relation with other metaproperties

Stronger metaproperties

• Intersection-closed subgroup property

Related metaproperties

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