18.62 (2014)

The spectrum of a finite group is the set of orders of its elements. Let $\omega$ be a finite set of positive integers. A group $G$ is said to be $\omega$-critical if the spectrum of $G$ coincides with $\omega$, but the spectrum of every proper section of $G$ is not equal to $\omega$.
$\qquad$ a) Does there exist a number $n$ such that for every finite simple group $G$ the number of $\omega(G)$-critical groups is less than $n$?
$\qquad$ b) For every finite simple group $G$, find all $\omega(G)$-critical groups.