The graph of atomic divisors and constructive recognition of finite simple groups

Abstract

The spectrum $\omega (G)$ of a finite group $G$ is the set of orders of elements of $G$. We present a polynomial-time algorithm that, given a finite set $\mathcal{M}$ of positive integers, outputs either an empty set or a finite simple group $G$. In the former case, there is no finite simple group $H$ with $\mathcal{M}=\omega (H)$, while in the latter case, $\mathcal{M}\subseteq \omega (G)$ and $\mathcal{M}\ne \omega (H)$ for all finite simple groups $H$ with $\omega (H)\ne \omega (G)$.