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Characteristicity is centralizer-closed
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 |
Statement
Property-theoretic statement
The subgroup property of being characteristic satisfies the subgroup metaproperty of being centralizer-closed.
Verbal statement
The centralizer of a characteristic subgroup is characteristic.
Statement with symbols
Suppose G is a group and H is a characteristic subgroup of G. Then, the centralizer CG(H) of H in G is also a characteristic 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.
