# Characteristicity is centralizer-closed

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) satisfying a subgroup metaproperty (i.e., centralizer-closed subgroup property)

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Get more facts about characteristic subgroup |Get facts that use property satisfaction of characteristic subgroup | Get facts that use property satisfaction of characteristic subgroup|Get more facts about centralizer-closed subgroup property

## 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 is a group and is a characteristic subgroup of . Then, the centralizer of in is also a characteristic subgroup of .

## 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.