2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/219300Suppose 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.Group Theory20D25 (Primary), 20D05, 20E28, 20E45 (Secondary)text