Characteristicity is centralizer-closed

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
|

Property "Page" (as page type) with input value "{{{property}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.Property "Page" (as page type) with input value "{{{metaproperty}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.

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 $C_G(H)$ 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.

Related facts

• Normality is centralizer-closed