If $G$ is a finite group whose order is divisible by a prime $p$ and $p^r$ is the maximal power of $p$ that divides it, the Sylow theorems tell us that the number $n_p$ of Sylow $p$-subgroups of $G$ is congruent to $1$ modulo $p$ and a divisor of $\lvert G\rvert/p^r$, and that if $n_p=1$ then $G$ is not simple. With this information alone we can decide that some numbers are not the order of a finite simple group.

How big is the set of numbers that these two facts exclude as candidates to be the order of a simple group?

By *big* here one can mean, say, the density in $\mathbb{N}$. One can also weigh numbers by the number of actual groups of each order, so as to turn the question into

What proportion of finite groups are known to be non-simple by using only these two Sylow theorems on the number of they Sylow subgroups?

Since most groups are apparently $p$-groups, the answer to this is probably one.

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