Normality is centralizer-closed

This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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Statement

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 $G$.

Related facts

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

Proof

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