A new solvability criterion for finite groups

TL;DR

This paper introduces a new criterion for finite group solvability based on conjugacy classes of prime power order elements, simplifying previous conditions and providing insights into the structure of simple groups.

Contribution

It establishes a weaker solvability criterion involving conjugacy classes and prime power elements, and proves a key property of nonabelian simple groups related to subgroup solvability.

Findings

Solvability can be determined by conjugacy classes of prime power order elements.

A key property of simple groups regarding non-solvable generated subgroups.

Extension of the criterion to families of groups closed under subgroups, quotients, and extensions.

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 consisting of elements of prime power order, there exist x in C and y in D with x and y generating a solvable group. 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 prime divisors a and b of |G| such that, for all elements x, y in G with |x|=a and |y|=b, the subgroup generated by x and y is not solvable. Further, using a recent result of Guralnick and Malle, we obtain a similar membership criterion for any family of finite groups closed under forming subgroups, quotients and extensions.