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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.

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Indeed. In fact $Y$ will be the quotient of $X$ by the group of deck transformations of $f$. (This is exactly like the case of a Galois extension of fields.) –  Zhen Lin Nov 29 '12 at 20:15
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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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