On control of the prime spectrum of the finite simple groups

The set $\pi(G)$ of all prime divisors of the order of a finite group $G$ is often called its prime spectrum. It is proved that every finite simple nonabelian group $G$ has sections $H_1,\dots,H_m$ of some special form such that $\pi(H_1)\cup\dots\cup\pi(H_m)=\pi(G)$ and $m\le5$, in the case when $G$ is an alternating or classical simple group, in addition, $m\le2$. Moreover, in any case, it is possible to choose the sections $H_i$ so that each of them is a simple nonabelian group, a Frobenius group, or (in one case) a dihedral group. If the above equality is realized for a finite group $G$, then we say that the set $\{H_1,\dots,H_m\}$ controls the prime spectrum of $G$. We also study some parameter $c(G)$ of finite groups $G$ related to the notion of control.