# Characteristicity is quotient-transitive

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

### Property-theoretic statement

The subgroup property of being a characteristic subgroup satisfies the subgroup metaproperty of being quotient-transitive.

### Statement with symbols

Suppose $H \le K \le G$ are subgroups such that $H$ is a characteristic subgroup of $G$, and $K/H$ is a characteristic subgroup of $G/H$. Then, $K$ is a characteristic subgroup of $G$.

## Proof

Given: A group $G$, subgroups $H \le K \le G$ such that $H$ is characteristic in $G$, and $K/H$ is characteristic in $G/H$

To prove: $K$ is characteristic in $G$

Proof: We pick any automorphism $\sigma$ of $G$, and want to show that $\sigma(K) = K$. For this, first observe that $\sigma(H) = H$, so $\sigma$ induces an automorphism on the quotient $G/H$, by the rule $gH \mapsto \sigma(g)H$. Call this automorphism $\sigma'$.

Then, $\sigma'$ is an automorphism of $G/H$. Since $K/H$ is characteristic in $G/H$, $\sigma'(K/H) = K/H$. Thus, for any $g \in K$, $\sigma'(gH) \in K/H$, and hence, unwrapping the definition, $\sigma(g) \in K$. Thus, $\sigma(K) \subset K$. Since the same holds for $\sigma^{-1}$, we conclude that $\sigma(K) = K$, completing the proof.