I am reading Giraud's notes (sorry they are in French) on Riemann surfaces and I have some questions about the end of the fifth section.
To begin with, a branched cover $f: X \to Y$ of degree $n$ (i.e. for every $y$ in $Y$ the sum over its fibre of the ramification degrees is constant n) is said to be Galois if $n$ is the order of the group Aut(X/Y). My first questions is: what is this group? In regular Galois theory one has a field extension and the automorphisms are those that fix the base field. If I had a surjective map or something I could see $X$ as an extension of $Y$ or something like that but that is not necessarily true because $X$ nor $Y$ may not be connected. In other (also French) notes I found that an automorphism of $f$ is an automorphism $\varphi \in$ Aut($X$) such that $f\circ \varphi = f$. And the group of automorphisms of $f$ is denoted Aut($X/Y$). This could be, but I find this notation quite confusing because $f$ is nowhere to be found at it. Maybe it does not depend on $f$? Meaning if I have another branched cover $g$ then they are equal.
Then, corollary $5.10$ says that every branched covering $f:X \to Y$ of degree $n$ is a quotient of a Galois branched covering, but I cannot even start dreaming about understanding it because the previous definitions are not clear to me.
I would find quite handy some suggested bibliography that proves this statement being similar to Giraud's notes but no so messy.
Thanks in advance