= Deck transformation group as a monodromy centralizer
Let $p:E\to X$ be a <connected> <covering space>, with $X$ <path-connected> and <locally path-connected>, and let $F=p^{-1}(x_0)$. Restriction to $F$ identifies the <deck transformation group> with the <centralizer> of the <monodromy action of a covering space> inside the <symmetric group> $\operatorname{Sym}(F)$.
A <deck transformation> commutes with lifting every <based loop>, so its restriction centralizes the <monodromy action of a covering space>. Conversely, let $\sigma$ be a <permutation> of $F$ that centralizes this action. Represent $e\in E$ as the endpoint of a lifted <path> $\gamma$ starting at $a\in F$, and send $e$ to the endpoint of the lift of $\gamma$ starting at $\sigma(a)$. Two such representations differ by a lifted <based loop>; commutation with its <monodromy> proves independence of the representation.
On a <path-connected> evenly covered neighborhood, this construction carries each sheet to a sheet by the corresponding local <homeomorphism>. It is therefore <continuous>. Applying the construction to $\sigma^{-1}$ gives its inverse, hence a <deck transformation>. Since $E$ is <path-connected>, a <deck transformation> is determined by one point, proving uniqueness and the asserted <group isomorphism>.
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