Centralizer-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

A subgroup property $\alpha$ is termed centralizer-closed if the following is true: whenever a subgroup $H$ of a group $G$ satisfies property $\alpha$, so does the centralizer $C_{G}(H)$.

Examples

Examples of subgroup properties that are centralizer-closed

	Quick phrase
Characteristic subgroup	invariant under all automorphisms automorphism-invariant strongly normal normal under outer automorphisms
Normal subgroup	invariant under inner automorphisms, self-conjugate subgroup same left and right cosets kernel of a homomorphism subgroup that is a union of conjugacy classes

Examples of subgroup properties that are not centralizer-closed

	Quick phrase
Subnormal subgroup