A new solvability criterion for finite groups

TL;DR

This paper introduces a new criterion for finite group solvability based on the existence of solvable generated subgroups from elements in conjugacy classes, simplifying previous conditions.

Contribution

It proves that a finite group is solvable if for all conjugacy classes, there exist elements generating a solvable subgroup, and establishes a key property of simple groups used in the proof.

Findings

Solvability can be determined by conjugacy class element pairs.

Finite simple groups have elements with orders that generate nonsolvable subgroups.

The new criterion simplifies checking solvability in finite groups.

Abstract

In 1968, John Thompson proved that a finite group $G$ is solvable if and only if every $2$-generator subgroup of $G$ is solvable. In this paper, we prove that solvability of a finite group $G$ is guaranteed by a seemingly weaker condition: $G$ is solvable if for all conjugacy classes $C$ and $D$ of $G$, \emph{there exist} $x\in C$ and $y\in D$ for which $\gen{x,y}$ is solvable. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if $G$ is a finite nonabelian simple group, then there exist two integers $a$ and $b$ which represent orders of elements in $G$ and for all elements $x,y\in G$ with $|x|=a$ and $|y|=b$, the subgroup $\gen{x,y}$ is nonsolvable.