= Simple group embedding from Sylow conjugation
A <finite nonabelian simple group> embeds in $A_{n_p}$, where $n_p$ is its number of <Sylow subgroups> for a prime dividing its order. It is not a <p-group> by the nontrivial-centre property. Thus its Sylow subgroups are proper and cannot be normal, so $n_p>1$. The conjugation action is nontrivial, and its normal <kernel of a group homomorphism> must therefore be trivial. The sign of this faithful permutation action must be trivial, since a nontrivial sign homomorphism would inject the group into a group of order two. <Lagrange's theorem> consequently gives $|G|\mid n_p!/2$.
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