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

## TL;DR

This paper introduces a polynomial-time algorithm to identify finite simple groups from a set of element orders, advancing the constructive recognition of such groups based on their spectra.

## Contribution

It presents a novel polynomial-time algorithm for recognizing finite simple groups from their spectra, specifically using the graph of atomic divisors.

## Key findings

- Algorithm correctly identifies simple groups from spectra
- Efficient recognition method for finite simple groups
- Provides a constructive approach to group recognition

## 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 $\mathcal{M}\neq\omega(H)$ for all finite simple groups $H$ with $\omega(H)\neq\omega(G)$.