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: $\S 64 \epsilon$