Thompson's conjecture on characterizing finite simple groups by order spectra

Let $G$ be a finite group and $S$ a finite simple group. Write $|G|$ for the order of $G$, and let $\pi_e(G)$ denote the set of element orders in $G$.

Thompson's conjecture. $G \cong S$ if and only if $|G|=|S|$ and $\pi_e(G)=\pi_e(S)$.

This conjecture was posed by the second author in a 1987 letter to J. G. Thompson. Based on previous work, it has been proven, so it is now a theorem.