Let $S$ be a finite simple group of Lie type and $p$ be a prime such that $|S|_p=p$. I want to get some restrictions on $S$ with the conditions that $S$ is generated by elements of order $p$ *and* the number of Sylow p-subgroups of S divides $(p-2)!$. I expect that there is no such a group.

On one hand I have some ideas how to use the condition on the number of Sylow p-subgroups. For instance, unless $S=PSL(2,p)$ we know that $p$ is not the defining characteristic of $S$, so every elements of order $p$ in $S$ is semisimple. Then I can use the information on the centralizers of semisimple elements in groups of Lie type.

On the other hand, I am not familiar with generation in simple groups of Lie type. I am wondering if we can obtain anything on $S$ and $p$ such that $|S|_p=p$ and $S$ is generated by elements of order $p$. Thanks.

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