Exercise 4.4.7 (If $H$ is the unique subgroup of a given order then $H$ is characteristic in $G$)

Proof. Suppose that $H$ is the unique subgroup of order $k$ in a group $G$. Let $\sigma \in \mathrm{Aut}(G)$. Then $|\sigma (H)|=|H|=k$, and $\sigma (H)$ is a subgroup of $G$. Since the only subgroup of $G$ is $H$, we obtain

This is true for every $\sigma \in \mathrm{Aut}(G)$, so $H$ is a characteristic subgroup of $G$. □