Characterization of groups with non-simple socle

TL;DR

This paper proves that for groups formed by the product of three copies of the group L_{2^m}(2) with m>5, the spectrum uniquely identifies the group among all finite groups.

Contribution

It establishes the spectral characterization of a specific class of finite groups formed by direct products of L_{2^m}(2) for m>5.

Findings

The spectrum uniquely determines the group in this class.

The result applies for all m>5.

Spectral characterization fails for smaller m.

Abstract

The spectrum of a finite group is a set of its element orders. We prove that if $m>5$ then the group $L_{2^m}(2)\times L_{2^m}(2)\times L_{2^m}(2)$ is uniquely determined by its spectrum in the class of finite groups