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From Groupprops

This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
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Contents 1 Statement 1.1 Property-theoretic statement 1.2 Verbal statement 1.3 Statement with symbols 2 Generalizations 3 Related facts 4 Proof

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.

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

• Characteristicity is centralizer-closed

Proof

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