Criterion of nonsolvability of a finite group and recognition of direct squares of simple groups

TL;DR

The paper establishes a spectral criterion for nonsolvability of finite groups and uniquely characterizes the direct square of Suzuki groups by spectrum, except for one specific case with four groups sharing the same spectrum.

Contribution

It introduces a new spectral criterion for nonsolvability and characterizes the spectrum of direct squares of Suzuki groups, identifying exceptions.

Findings

A spectral criterion for nonsolvability based on prime divisors and element orders.

Unique spectral characterization of $Sz(q) imes Sz(q)$ for most $q eq 32$.

Identification of four groups sharing the spectrum with $Sz(32) imes Sz(32)$.

Abstract

The spectrum $\omega(G)$ of a finite group $G$ is the set of orders of its elements. The following sufficient criterion of nonsolvability is proved: if among the prime divisors of the order of a group $G$, there are four different primes such that $\omega(G)$ contains all their pairwise products but not a product of any three of these numbers, then $G$ is nonsolvable. Using this result, we show that for $q\geqslant 8$ and $q\neq 32$, the direct square $Sz(q)\times Sz(q)$ of the simple exceptional Suzuki group $Sz(q)$ is uniquely characterized by its spectrum in the class of finite groups, while for $Sz(32)\times Sz(32)$, there are exactly four finite groups with the same spectrum.