Let $\pi (k)$ be the set of prime divisors of $k$ and let $\pi (G)=\pi (|G|)$. Let $G$ be finite simple group with $\pi (G)\subseteq \pi (p^{2}-1)$, where $p$ is prime.
I would like to know is there any classifications for group $G$?
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Do you mean for any prime $p$ or a specific prime $p$? – Alexander Gruber Feb 24 '13 at 20:05
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@AlexanderGruber: $p$ is prime but not Mersenne or Fermat. Thanks. – user2132 Feb 26 '13 at 13:15
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$G$ could be any finite simple group. For any positive integer $n$, there exists a prime $p$ congruent to $1$ mod $n$ by Dirichlet's theorem. And of course if $p \equiv 1 \pmod n$, then every divisor of $n$ is a divisor of $p-1$ and hence of $p^2-1$.
Andreas Caranti
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Chris Eagle
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