Finite groups isospectral to simple groups

TL;DR

This paper surveys the recognition of finite simple groups based on their spectra, focusing on identifying which simple groups can be uniquely determined by the set of element orders.

Contribution

It compiles and reviews known results on recognizing finite simple groups by spectrum, highlighting which cases are fully solved.

Findings

Complete list of simple groups with solved recognition problem

Summary of techniques used for spectrum-based recognition

Identification of open problems in the area

Abstract

The spectrum of a finite group is the set of element orders of this group. The main goal of this paper is to survey results concerning recognition of finite simple groups by spectrum, in particular, to list all finite simple groups for which the recognition problem is solved.