A new construction of Bol loops of order 8 k

Abstract

One of the most interesting constructions of nonassociative Moufang loops is the construction by the first author of the loops M ( G , 2 ) [Trans. Amer. Math. Soc. 188 (1974) 31–51], in which the new loop is constructed as a split extension of a nonabelian group G by a cyclic group of order 2. This same construction will not produce non-Moufang Bol loops, even if we start with a Moufang loop instead of a group. We generalize the construction to produce a large class of Bol loops as extensions of B by C m × C n , where B can be any group or even any loop which satisfies the right Bol identity. We determine conditions under which the newly constructed loops will be Moufang and when they will be associative. We also find the nuclei, centrum, centre, and commutator/associator subloop of a loop constructed by this method, and we investigate which known Bol loops of small orders arise in this way.