A solvable version of the Baer--Suzuki Theorem

TL;DR

This paper characterizes when elements of prime order greater than 3 are contained in the solvable radical of a finite group, providing a criterion involving conjugates and solvability, and explores exceptions for smaller primes.

Contribution

It establishes a solvability criterion for elements of prime order in finite groups and identifies exceptions for prime 3, extending the Baer--Suzuki theorem.

Findings

Elements of prime order p > 3 are in the solvable radical iff their conjugates generate solvable subgroups.

For almost simple groups, such elements do not always generate solvable subgroups, with explicit exceptions for p=3.

The paper explicitly describes the few cases where the criterion does not hold for p=3.

Abstract

Suppose that G is a finite group and x in G has prime order p > 3. Then x is contained in the solvable radical of G if (and only if) <x,x^g> is solvable for all g in G. If G is an almost simple group and x in G has prime order p > 3 then this implies that there exists g in G such that <x,x^g> is not solvable. In fact, this is also true when p=3 with very few exceptions, which are described explicitly.