Characteristicity is centralizer-closed

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