Centralizer-closed subgroup property

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This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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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 of subgroup properties that are centralizer-closed

 Quick phrase
Automorph-conjugate subgroup
C-closed normal subgroup
Characteristic subgroupinvariant under all automorphisms
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