On Bruck Loops of 2-power Exponent

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 [BS3] and in [S]. Moreover, we determine the groups associated to Bruck loops of 2-power exponent under the assumption that every nonabelian simple group $S$ is either passive or isomorphic to ${\text{PSL}}_2(q)$, $q-1\ge 4$ a $2$-power. In a separate paper it is proven that indeed every nonabelian simple group $S$ is either passive or isomorphic to ${\text{PSL}}_2(q)$, $q-1\ge 4$ a $2$-power [S]. The results obtained here are used in [BS3], where we determine the structure of the groups associated to the finite Bruck loops.