TITLE: A good way to understand Galois covering?
QUESTION [17 upvotes]: A covering map $f:X\rightarrow Y$ is called Galois if for each $y\in Y$ and each pair of lifts $x, x^{'}$, there is a covering transformation taking $x$ to $x^{'}$. What is a good way to understand this definition? It seems to me that $f$ is Galois if and only if $Y$ is obtained from $X$ as a quotient of some group.

REPLY [14 votes]: In the setting of (complex) algebraic geometry, the covering is Galois if and only if the function field $K(X)$ is a Galois extension of the function field $K(Y)$. Moreover, if $f$ is Galois, then the Galois group of the extension is exactly the deck transformation group $G$. As you've already noticed.  If $f$ is Galois, then $Y$ is isomorphic to $X/G$, where $G$ is the Galois group.
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