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 $L_{2}(p)$?

  • $\begingroup$ I have been assuming that $p$ is prime. Is that right? $\endgroup$ – Derek Holt Apr 22 '12 at 11:09
  • $\begingroup$ 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 K Apr 22 '12 at 13:00
  • 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 Holt Apr 23 '12 at 9:14

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.