Characterizations of the Solvable Radical

TL;DR

This paper establishes a bound on the number of elements needed from a conjugacy class in a finite group to ensure the generated subgroup is solvable, with explicit bounds derived using classification and alternative methods.

Contribution

It proves the existence of a universal constant k such that a conjugacy class with certain solvability properties generates a solvable subgroup, providing explicit bounds with and without classification.

Findings

For k=4, the result holds using the Classification of Finite Simple Groups.

A direct proof without classification gives k=10.

A slightly modified argument reduces k to 7.

Abstract

We prove that there exists a constant $k$ with the property: if $\calC$ is a conjugacy class of a finite group $G$ such that every $k$ elements of $\calC$\ generate a solvable subgroup then $\calC$ generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take $k=4$. We also present proofs that do not use the Classification theorem. The most direct proof gives a value of $k=10$. By lengthening one of our arguments slightly, we obtain a value of $k=7$.