A famous result of Galois, in his letter to Auguste Chevalier, is that for $p$ prime $>11$ the group $\operatorname{PSL}(2,\mathbb{F}_p) $ does not embed in the symmetric group $\mathfrak{S}_p$. The standard proof nowadays goes through the classification of subgroups of $\operatorname{PSL}(2,\mathbb{F}_p) $ (Dickson's theorem), which is far from trivial. Does anyone know a simpler argument?