I'm trying to prove that if $G$ is a simple group and $H$ is a proper subgroup of $G$ of index $n$ then $G$ is isomorphic to a subgroup of $S_n$.
To do this, I've been trying to find a homomorphism between $G$ and permutations on the set of right cosets of $H$, because this would give me a subgroup of $S_n$, but I've failed to find one with a trivial kernel(or at least one I can prove has a trivial kernel). Nor am I sure how to use the condition that $G$ is simple. I'm not even convinced this is the right approach, but I can't think of another way to get permutations on $n$ things.
Thanks!
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