We know all 2transitive simple groups by Dixon's book (Permutation groups). Now let $G$ be finite simple group $2$transitive and $p(p^{2}1)/2$ divides order $G$ and also $\pi (G)\subseteq \pi (p(p^{2}1))$. Is it true $G$ isomorphic to $L_{2}(p)$?
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If I have understood the question correctly, then there seem to be lots of small counterexamples, such as $G=A_6, p=5$; $G=L_2(8)$ or $U_3(3)$, $p=7$; $G=M_{11}$ or $M_{12}$, $p=11$. Added later: I thought of two more examples: $G=L_2(27), p=13$ and $G=L_3(5), p=31$. The interesting question is whether there are only finitely many examples. I would guess yes, but it could be hard to prove it. 

