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
View a complete list of subgroup metaproperties
View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

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
Central factorfactor in central product
product with centralizer is whole group
quotient action by outer automorphisms is trivial
every inner automorphism restricts to an inner automorphism
Characteristic subgroupinvariant 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
Local powering-invariant subgroup
Normal subgroupinvariant under inner automorphisms, self-conjugate subgroup
same left and right cosets
kernel of a homomorphism
subgroup that is a union of conjugacy classes
Powering-invariant subgroup

Examples of subgroup properties that are not centralizer-closed

 Quick phrase
Subnormal subgroup
Transitively normal subgroup