Normality is centralizerclosed
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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Contents
Statement
Propertytheoretic statement
The subgroup property of being normal satisfies the subgroup metaproperty of being centralizerclosed.
Verbal statement
The centralizer of a normal subgroup is normal.
Statement with symbols
Suppose is a group and is a normal subgroup of . Then, the centralizer of in is also a normal subgroup of .
Related facts
Generalizations
Autoinvariance implies centralizerclosed: 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 centralizerclosed
 monomial automorphisminvariance is centralizerclosed.
Analogues for Lie rings
 Invariance under any set of derivations is centralizerclosed
 Ideal property is centralizerclosed
 Derivationinvariance is centralizerclosed