= A simple group with an index-n subgroup embeds in the alternating group
{title2=$n\geq3\implies G\hookrightarrow A_n$}
For a finite <simple group> with a <subgroup> of index $n\geq3$, the <coset action> is nontrivial and its <kernel> is a <normal subgroup>, hence trivial. Its <sign homomorphism> must also be trivial: otherwise simplicity would give an isomorphism to $C_2$, which cannot act transitively on at least three points. Thus the <image> lies in the <alternating group>. In particular <Lagrange's theorem> gives $|G|\mid n!/2$. This can exclude simplicity with very little counting: a group of order $112$ has a <subgroup> of index seven by <Sylow theorems>, but $112$ does not divide $7!/2$.
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