# Characteristicity is transitive

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This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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## Statement

### Property-theoretic statement

The subgroup property of being characteristic satisfies the subgroup metaproperty of being transitive.

### Verbal statement

A characteristic subgroup of a characteristic subgroup is characteristic in the whole group.

### Symbolic statement

Let $H$ be a characteristic subgroup of $K$, and $K$ a characteristic subgroup of $G$. Then, $H$ is a characteristic subgroup of $G$.

## Related facts

### Generalizations

Balanced implies transitive: Any subgroup property that can be expressed as a balanced subgroup property is transitive. Characteristicity is a special case. Other special cases include: