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  1. Every etale morphism is finite over some nonempty open set. (For instance, locally somewhere write it as a standard etale morphism $(A[t]/f(t))_{g(t)}$, then consider the open set where the norm of $g$, that is, the resultant of $f$ and $g$, is nonzero.) If a group homomorphism $H \to G$ is finite over some nonempty open set, it is finite over all translates of that set, so it is finite everywhere.

  2. It is isomorphic to $\pi_1(G,\overline{1})$. In general, if $X \to Y$ is finite etale and Galois, the kernel of the map $\pi_1(Y) \to Gal(X/Y)$ is $\pi_1(X)$.2.\pi_1(X)$.
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  1. Every etale morphism is finite over some nonempty open set. (For instance, locally somewhere write it as a standard etale morphism $(A[t]/f(t))_{g(t)}$, then consider the open set where the norm of $g$, that is, the resultant of $f$ and $g$, is nonzero.) If a group homomorphism $H \to G$ is finite over some nonempty open set, it is finite over all translates of that set, so it is finite everywhere.

  2. It is isomorphic to $\pi_1(G,\overline{1})$. In general, if $X \to Y$ is finite etale and Galois, the kernel of the map $\pi_1(Y) \to Gal(X/Y)$ is $\pi_1(X)$.2.