J.D.Phillips, Finite Bruck loops

Abstract. Bruck loops are Bol loops satisfying the automorphic inverse prop-erty. We prove a structure theorem for finite Bruck loops X, showing that X is essentially the direct product of a Bruck loop of odd order with a 2-element Bruck loop. The former class of loops is well understood. We identify the minimal obstructions to the conjecture that all finite 2-element Bruck loops are 2-loops, leaving open the question of whether such obstructions actually exist. Let X be a magma; that is, X is a set together with a binary operation ◦ on X. For each x ∈ X we obtain maps R(x) and L(x) on X defined by R(x) : y → y ◦ x and L(x) : y → x ◦ y called right and left translation by x, respectively. A loop is a magma X with an identity 1 such that R(x) and L(x) are permutations of X for all x ∈ X. In essence loops are groups without the associative axiom. See [Br] for further discussion of basic properties of loops. Certain classes of loops have received special attention: A loop X is a (right) Bol loop if it satisfies the (right) Bol identity (Bol) for all x, y, z ∈ X