An old qual question on Sylow $p$-groups:
Assume that $p$ is an odd prime and $G$ is a finite simple group with exactly $2p+1$ Sylow $p$-groups. Prove that the Sylow $p$-groups of $G$ are abelian.
I am not sure which technique to apply to show that these groups are abelian. I know that the $2p+1$ Sylow $p$-subgroups are conjugate, but this is a statement about the entire sets rather than their individual elements and how they interact with each other. For any Sylow $p$-subgroup $P$ we have $\vert G/P\vert $=$\vert G\vert / p$.
I am not even sure if Sylow's theorem is needed since we are already given that $n_p=2p+1$. How can I proceed?