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:

$\phi \left({H}\right) = H$

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:

$\phi \restriction_H \left({K}\right) = K$

but this immediately implies that:

$\phi \left({K}\right) = K$

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


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

$\blacksquare$


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