2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/145913In the spirit of Glauberman's fundamental work in B-loops and Moufang loops, we prove Cauchy and strong Lagrange theorems for Bol loops of odd order. We also establish necessary conditions for the existence of a simple Bol loop of odd order, conditions which should be useful in the development of a Feit-Thompson theorem for Bol loops. Bol loops are closely related to Aschbacher's twisted subgroups, and we survey the latter in some detail, especially with regard to the so-called Aschbacher radical.20 pages, AMS-LaTeX, to appear in Rocky Mountain Math. J. Version 2 corrects some serious mistakes, and includes many cosmetic changes suggested by the refereeGroup Theory20N05text