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Suppose we have a topological space [itex]X[/itex], a point [itex]x\in X[/itex] and a homomorphism [itex]\rho:\pi(X,x) \rightarrow S_n[/itex] with transitive image. Consider the subgroup [itex]H[/itex] of [itex]\pi(X,x)[/itex] consisting of those homotopy classes [itex][\gamma][/itex] such that [itex]\rho([\gamma])[/itex] fixes the index [itex]1\in \{1,\ldots,n\}[/itex]. I know that [itex]H[/itex] induces a covering space [itex]p:Y\rightarrow X[/itex]. However, I can't understand why the monodromy map of [itex]p[/itex] is exactly [itex]\rho[/itex].

Can anyone help me?