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 $\mathrm{Sym}(F)$.

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.