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

Property-theoretic statement

The subgroup property of being characteristic satisfies the subgroup metaproperty of being centralizer-closed.

Verbal statement

The centralizer of a characteristic subgroup is characteristic.

Statement with symbols

Suppose $G$ is a group and $H$ is a characteristic subgroup of $G$. Then, the centralizer $C_G(H)$ of $H$ in $G$ is also a characteristic subgroup of $G$.

Generalizations

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.

Related facts

• Normality is centralizer-closed