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Let, be $V$ a connected smooth manifold and $q_1,q_2\in V$ and $F:U\to V$ a connected covering of degree $d$. This covering induces two monodromy representations $\rho_1:\pi_1(V,q_1)\to S_d $ and $\rho_2:\pi_1(V,q_2)\to S_d $ where $S_d$ denotes the symmetric group of all permutations on $d$ indices. How to show that $\rho_1$ and $\rho_2$ they are isomorphic, tha is, exists isomrphims $\phi:\pi_1(V,q_1)\to \pi_1(V,q_2)$ and $ \psi:S_d\to S_d$, such that $\rho_1\circ\psi = \phi\circ\rho_2. $

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take a path connecting $p_1$ and $p_2$. This defines an isomorphism between the fundamental group naturally. Also defines a bijection between the fibers,and this induces a isomorphism among symmetric groups. The isomorphisms satisfies the desired.