A solvable version of the Baer--Suzuki Theorem
Group Theory
2009-02-11 v1
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.
Cite
@article{arxiv.0902.1738,
title = {A solvable version of the Baer--Suzuki Theorem},
author = {Simon Guest},
journal= {arXiv preprint arXiv:0902.1738},
year = {2009}
}