I am self studying Fields and Galois Theory from Algebra by Thomas Hungerford and I have a question in this theorem s proof.
I have question in line 1 where author writes "In any case $G$ is a subgroup of $\operatorname{Aut}_K F $ .
Definition of $\operatorname{Aut}_K F $ is group of all $K$-automorphisms of FF , where $G$ is a group of automorphisms of $F$.
So, I think since conditions are on $\operatorname{Aut}_K F $ that it must be a $K$-module homomorphism , so $G$ must be a larger set $\operatorname{Aut}_K F$ , but the opposite is given. So, can anyone please explain why opposite is given.