Let $K \leq E \leq F$ be fields such that $[E:K]=n$. Let $Aut_K F$ act on the set $S$ of intermediate fields where $\sigma(I)$ gives the action of $\sigma \in Aut_K F$ on an intermediate field $I$. Show that the orbit of $E$ has at most $n$ elements.
Is it really obvious that $Aut_K F$ is a subset of the stabilizer of $E$? My intuition fails me here.