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
Automorph-conjugate subgroup
C-closed normal subgroup
Characteristic subgroup	invariant under all automorphisms automorphism-invariant strongly normal normal under outer automorphisms
Characteristic subgroup of group of prime power order
Finite direct power-closed characteristic subgroup

Examples of subgroup properties that are not centralizer-closed

	Quick phrase
Subnormal subgroup
Transitively normal subgroup