On 2-Engel groups and Bruck loops

Abstract

Abstract. A classical construction associates a Bruck loop with a Moufang or Bol loop on which the squaring map is a permutation. The idea of that construction is now extended to a quadratically closed Bol loop or group (where the squaring operation is surjective) that satisfies the 2-Engel condition (for groups: elements commute with their conjugates). The Bruck loop is built on a space of sequences of elements from the original Bol loop. More generally, given a quadratically closed loop with the 2-Engel condition from the join of the varieties of diassociative and (right) Bol loops, a power-associative loop with the right inverse and automorphic inverse properties is constructed in similar fashion.