We have a connected topological space $X$ and $p:E\to X$ a Galois covering projection with $E$ not necessarily connected, let us call $\text{Aut}(E/X)=G$. Suppose it is given that every homomorphism of $\pi_1(X,x)\to G$ is trivial. Then we have to show that $p$ is isomorphic to the trivial projection $X\times G\to X$ in the category of coverings of $X$. We may assume $X$ has a universal covering space which is simply connected.
What I tried to do was pick $x\in X$ and then pick something in the preimage, say $y$. If we restrict $p$ to the connected component $C$ of $E$ containing $y$(is it still a covering projection? I suspect it is.), then $p$ induces an injective map between the fundamental groups of $X$ and $C$. But I don't know how to use the rest of the information here. I mean I can apply the Galois correspondence to $C$ now and get $\text{Aut}(C/X)$ is a subgroup of $\pi_1(X,x)$. But how to proceed from here?
