We continue the work of Aschbacher, Kinyon and Phillips (2006) [AKP06] as well as of Glauberman (1964Glauberman ( , 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 PSL 2 (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.

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The goal of this paper is two-fold. First we provide the information needed to study Bol, A r or Bruck loops by applying group theoretic methods. This information is used in this paper as well as in Baumeister and Stein (2010) [BS3] and in Stein (2009) Moreover, we determine the groups associated to Bruck loops of 2-power exponent under the assumption that every non-abelian simple group S is either passive or isomorphic to PSL 2 (q), q − 1 4 a 2-power. In a separate paper it is proven that indeed every nonabelian simple group S is either passive or isomorphic to PSL 2 (q), q − 1 4 a 2-power (Stein, 2009) [S]. The results obtained here are used in Baumeister and Stein (2010) [BS3], where we determine the structure of the groups associated to the finite Bruck loops.

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