# characteristic group

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