We know all 2-transitive 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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$\begingroup$ I have been assuming that $p$ is prime. Is that right? $\endgroup$– Derek HoltCommented Apr 22, 2012 at 11:09
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$\begingroup$ Thank you for you answer. Yes $p$ is prime. Also let the number of Sylow $p$-subgroup $G$ equal to the number of Sylow $p$-subgroup $PSL(2,p)$. Now: whether $G$ isomorphic to $PSL(2,p)$? $\endgroup$– R KCommented Apr 22, 2012 at 13:00
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2$\begingroup$ If $G$ has $p+1$ Sylow $p$-subgroups, then its action by conjugation on the set of Sylow $p$-subgroups is 2-transitive, and the point stabilizer has order $p(p-1)/2$ with a normal subgroup of order $p$. A group with those properties is isomorphic to ${\rm PSL}(2,p)$ - you don't even need the classification of fintie simple groups for that - it follows from the result proved in: Hering, Christoph; Kantor, William M.; Seitz, Gary M. Finite groups with a split BN-pair of rank 1. I. J. Algebra 20 (1972), 435–475. $\endgroup$– Derek HoltCommented Apr 23, 2012 at 9:14
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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.
answered Apr 21, 2012 at 18:37