A proper index-$n$ subgroup of a simple group yields a faithful action on $n$ cosets.Suggest a slogan
Let \(G\) be a simple group and \(H < G\) a proper subgroup of finite index \([G:H] = n\). Then there is an injective group homomorphism \(\phi\colon G \injects S_n\).
The action of \(G\) on the set \(G/H\) of left cosets by \(g \cdot xH \coloneqq gxH\) gives a homomorphism \(\phi\colon G \to \operatorname{Sym}(G/H) \cong S_n\). If \(g \in \ker\phi\), then \(gH = H\), so \(\ker\phi \subseteq H \neq G\). Since \(\ker\phi\) is a normal subgroup of the simple group \(G\) and \(\ker\phi \neq G\), \(\ker\phi\) is trivial.