Characteristicity is centralizer-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) satisfying a subgroup metaproperty (i.e., centralizer-closed subgroup property)
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Statement with symbols
Auto-invariance implies centralizer-closed: Any subgroup property that can be described as the invariance property with respect to a certain automorphism property, is closed under taking centralizers.