Abstract
In the spirit of Glauberman’s fundamental work in B-loops and Moufang loops [18, 19], 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 [1], and we survey the latter in some detail, especially with regard to the so-called Aschbacher radical.
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