Let $G$ be a group, $M$ be a maximal subgroup of $G$ and $\alpha \in \operatorname{Aut}(G)$. I want to show that $\alpha(M)$ is a maximal subgroup of $G$.
I know $\alpha(M)= \lbrace \alpha(m) \mid m \in M \rbrace$. Suppose contrary that $\alpha(M) <K<G$. please help me to complete this proof.
$M<\alpha^{-1}(K)<G$, but $M$ is supposed to be maximal.
– Ángel Mario Gallegos Aug 20 '18 at 06:52