Generation in finite simple groups of Lie type 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.
 A: You can easily find many examples of such groups using GAP. The following short program does it (it skips over the numerous examples of type $A_1$, and I did not bother to filter out alternating and sporadic examples)
it:=SimpleGroupsIterator();
for G in it do
    if IsPSL(G) and ParametersOfGroupViewedAsPSL(G)[1] = 2 then continue; fi;
    n := Size(G);
    primes := Set(Factors(n));
    for p in primes do
        if n mod p^2 = 0 then continue; fi;
        N := Normalizer(G, SylowSubgroup(G, p));
        numSyl := n / Size(N);
        if Factorial(p-2) mod numSyl = 0 then
            Print(G, "\n");
            break;
        fi;
    od;
od;

Running this in GAP 4.8.4, I get this list (note that it lists $PSL(2,7)$, even though the code should skip $PSL(2,p)$ -- I think that's because the undocumented attribute ParametersOfGroupViewedAsPSL I used detects it as $PSL(3,2)$, which is of course isomorphic)  :
PSL(2,7)
A7
PSL(3,3)
M11
Sz(8)
PSU(3,4)
M12
J_1
PSL(3,5)
M22
PSp(4,4)
PSU(3,8)
PSU(3,7)
PSL(5,2)
M23
PSL(3,8)
A11
Sz(32)
PSU(3,9)
J3
PSU(3,11)
O-(8,2)
M24
PSL(3,13)
PSU(3,13)
PSL(4,4)
PSU(4,4)
PSL(3,16)
PSp(4,9)
A13
PSU(3,16)
PSL(3,19)
G_2(5)
PSL(3,17)
PSL(4,5)
Ree(27)
PSp(4,11)
PSL(6,2)
Sz(128)
PSL(3,25)
PSL(3,23)
Ru
PSU(3,25)
PSU(3,29)
PSU(5,3)
PSL(3,27)
PSU(3,27)
PSL(3,31)
PSU(3,32)

The computation would go on, but I aborted it at this point.
