A new construction of Bol loops of order 8k

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.  2005 Elsevier Inc. All rights reserved.

Schreier loops

“…, n − 1, yields a loop L(Id, f). This loop is according to Theorem 2.3 and Lemma 6.1 in [6] a right Bol loop but it is not Moufang. Let N be the group of order 2 and let K be the elementary abelian group of order 8.…”

On the isotopic characterizations of generalized Bol loops

“…Some later results on Bol loops and Bruck loops can be found in Bruck [9], Solarin [41], Adén íran and Akinleye [2], Bruck [10], Burn [11], Gerrit Bol [8], Blaschke and Bol [7], Sharma [32,33], Adén íran and Solarin [4]. In the 1980s, the study and construction of finite Bol loops caught the attention of many researchers among which are Burn [11,12,13], Solarin and Sharma [34,37,38,39] and others like Chein and Goodaire [14,15,16], Foguel at. al.…”

Bol Loops of Nilpotence Class Two

“…This observation forms the basis for the construction of a new class of Bol loops. The construction in the case that r 2 = s 2 = t 2 = 1 is described in [CG1].…”