PSL(2,q) has order q^3 - q = q(q-1)(q+1).
If q is a power of two, then 2-Sylow subgroup will not be largest if q+1 is prime, but it is equivalent to q+1 is a Fermat prime.
I define NSG(n) as PSG(2,2^2^n). Therefore NSG(n) will be simple iff 2^2^n+1 is Fermat prime.
Question: are there only five Fermat primes?