Why do Automorphisms map maximal subgroups to maximal subgroups?

Question

I am trying to prove that $\Phi(G)$ is characteristic. However, I can't immediately see why all maximal subgroups are mapped to maximal subgroups, which I believe is an essential ingredient in the proof.

Answer 1

Just try to see the fact that, if $\phi: G\to G$ is an automorphism of groups, then for any subgroup $K$ of $G$, $\phi^{-1}(K)$ is a subgroup of $G$.

Assume $H$ is a maximal subgroup of $G$ (Domain of $\phi$) then if there exists subgroup $K$ of $G$ (Image of $\phi$) s.t. $\phi (H)\subset K$ then we see $H\subset \phi^{-1}(K)$ contradicting the maximality of $H$. So $\phi(H)$ is maximal subgroup of $G$.

Answer 2

Let $\phi:G \rightarrow G \in Aut(G)$. It is easy to show with subgroup criterion that $\phi$ maps a subgroup to another subgroup. As automorphism is an isomoprhism, it would map any subgroup to another subgroup with the same order.

Assume H is a maximal subgroup of G. Assume there exists $K < G$ such that $\phi(H) < K$. Note that as automorphism maps subgroup to another subgroup, so is $\phi^{-1}(K)$ also a subgroup. As $\phi(H) < K$, there are elements subset of $K$ that their domain is exactly $H$, so we have $H < \phi^{-1}(K) < G$. This contradicts $H$ being a maximal subgroup of G.