# Characteristic Subgroup is Transitive

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## Theorem

Let $G$ be a group.

Let $H$ be a characteristic subgroup of $G$.

Let $K$ be a characteristic subgroup of $H$.

Then $K$ is a characteristic subgroup of $G$.

## Proof

Let $\phi: G \to G$ be a group automorphism.

Since $H$ is a characteristic subgroup of $G$, we have:

- $\phi \sqbrk H = H$

Thus, from Group Homomorphism Preserves Subgroups, $\phi {\restriction_H}$, the restriction of $\phi$ to $H$, is an automorphism of $H$.

Now since $K$ is a characteristic subgroup of $H$, we have that:

- $\phi {\restriction_H} \sqbrk K = K$

but this immediately implies that:

- $\phi \sqbrk K = K$

by definition of the restriction $\phi {\restriction_H}$.

That is, $K$ is a characteristic subgroup of $G$ as well.

$\blacksquare$

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 64 \epsilon$