Let$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$Let $\pi$ be a finite set of primes. A finite group $G$ is a $\pi$-group if all primes dividing $|G|$ lie in $\pi$. Is it true that there are only finitely many isomorphism classes of simple $\pi$-groups?
We could try to approach this using the classification, as follows. There are only finitely many sporadic groups, so we can ignore them. It is clear that there are only finitely many prime cyclic groups or alternating groups that are $\pi$-groups. For $SL(n,p^v)$$\SL(n,p^v)$ to be a $\pi$-group we must have $p\in\pi$, and $n$ must be less than the smallest prime outside $\pi$, so there are only finitely many possibilities for the pair $(p,n)$. It is not so obvious what to say about $v$, although the critical case seems to be when $v$ is also prime, and experiment suggests that the largest prime factor of $|SL(n,p^v)|$$|\SL(n,p^v)|$ grows very rapidly with $v$ in that case. The story should be similar for the simple quotient $PSL(n,p^v)$$\PSL(n,p^v)$ and for other families of Lie type. However, it would be more satisfactory to have a proof independent of the classification; I do not know whether that is at all plausible.
