  In 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.
