The finite Bruck loops

We continue the work of Aschbacher, Kinyon and Phillips (2006) [AKP06], as well as of Glauberman (1964, 1968) [G64,G68] by describing the structure of the finite Bruck loops. We show that a finite Bruck loop X is the direct product of a Bruck loop of odd order with either a soluble Bruck loop of 2-power order or a product of loops related to the groups PSL2(q), q=9 or q⩾5 a Fermat prime. The latter possibility does occur as is shown in Nagy (2008) [N08], and Baumeister and Stein (2010) [BS10]. As corollaries we obtain versions of Sylow's, Lagrange's and Hall's Theorems for loops.