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For a path connected topological space $X$, the fundamental groups at two different base points are isomorphic.

Does the same hold for the monodromy group of a covering map $Y \rightarrow X$, when $X$ is path connected?

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By monodromy you mean the quotient $\pi_1(X,x)/p_*\pi_1(Y,y)$? Then you should at least assume that this is a group. Moreover if Y is not connected then this would obviously be wrong.. If it is connected then both groups turning up are isomorphic by your observation, and $p_*$ is always injective which should prove your claim. –  mland Jun 2 '12 at 7:29
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