3-Abelian Groups and Commutative Moufang Loops

Abstract

It has been recently shown there exist only two non-abelian commutative Moufang loops of order 81 = 3 4 , and that the number v s of pairwise non-isomorphic exponent 3 commutative Moufang loops (resp. distributive symmetric quasigroups) whose 3-order is s satisfies v 4 = 2 = v 5 and v 6 = 4. The object of this note is to establish a connection between these results and some theorems of classification of 3-abelian groups [i.e. the groups whose law satisfies the identical relation x 3 y 3 = ( xy ) 3 ]. We use techniques introduced by Bruck so as to obtain group-theoretical descriptions of the considered quasigroups.