If H is the unique subgroup of a given order in a group G, prove H is characteristic in G.
Let $\displaystyle \theta$ be an automorphism of $\displaystyle G$ then $\displaystyle |\theta (H)| = |H|$ - why? But $\displaystyle \phi (H)$ is a subgroup of $\displaystyle H$ because homomorphism images of subgroups are subgroups. And by uniqueness we have $\displaystyle \phi (H) = H$. Therefore, $\displaystyle H$ is invariant under all automorphism i.e. it is a characteristic subgroup.