Normality is centralizer-closed
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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Contents
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 is a group and
is a normal subgroup of
. Then, the centralizer
of
in
is also a normal subgroup of
.
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