Nonassociative triples in involutory loops and in loops of small order

Abstract

A loop of order n possesses at least 3n(2) - 3n + 1 associative triples. However, no loop of order n > 1 that achieves this bound seems to be known.

If the loop is involutory, then it possesses at least 3n(2) - 2n associative triples. Involutory loops with 3n(2) - 2n associative triples can be obtained by prolongation of certain maximally nonassociative quasigroups whenever n-1 is a prime greater than or equal to 13 or n-1 = p(2k), p an odd prime.

For orders n <= 9 the minimum number of associative triples is reported for both general and involutory loops, and the structure of the corresponding loops is described.

Keywords

• quasigroup
• loop
• prolongation
• involutory loop
• associative triple
• maximally nonassociative