There's this theorem in my book that says that if $K/F$ is normal extension and $H≤Aut(K/F)$ then $ord(H)=[K:K_H]$ ($K_H$ stands for fixed field by $H$)
But as I read through the proof of it, it seemed that the proof was not at all using the conditions that $K/F$ is normal extension and $H$ is subgroup of $Aut(K/F)$.
I wonder if I had read the proof wrong or if indeed the theorem is still true without those conditions - thus that for arbitrary finite group $G$ of automorphisms of $K$, $ord(G)=[K:K_G]$