Normality is centralizer-closed

Property-theoretic statement
The subgroup property of being normal satisfies the subgroup metaproperty of being centralizer-closed.

Verbal statement
The centralizer of a normal subgroup is normal.

Statement with symbols
Suppose $$G$$ is a group and $$H$$ is a normal subgroup of $$G$$. Then, the centralizer $$C_G(H)$$ of $$H$$ in $$G$$ is also a normal 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. Some other instances of this generalization are:


 * Characteristicity is centralizer-closed
 * monomial automorphism-invariance is centralizer-closed.

Analogues for Lie rings

 * Invariance under any set of derivations is centralizer-closed
 * Ideal property is centralizer-closed
 * Derivation-invariance is centralizer-closed