Assume that $\displaystyle p$ is an odd prime and $\displaystyle G$ is a finite simple group with exactly $\displaystyle 2p+1$ Sylow $\displaystyle p$-subgroups. Prove that the Sylow $\displaystyle p$-subgroups are abelian.

I haven't made much progress: It's easy if $\displaystyle p^2$ doesn't divide the order of $\displaystyle G$, since then the Sylow $\displaystyle p$-subgroups have order $\displaystyle p$ and are therefore cyclic.

Thanks,

Hollywood