Few associative triples, isotopisms and groups

Abstract

Let Q be a quasigroup. For let be the principal isotope.

Put and assume that. Then , and for every there is , where.

If G is a group and is an orthomorphism, then for every. A detailed case study of is made for the situation when , and both and are "natural" near-orthomorphisms.

Asymptotically, if G is an abelian group of order n. Computational results: and , where.

There are also determined minimum values for a(G(alpha,beta)), G a group of order <= 8.

Keywords

• Quasigroup
• Associative triple
• Associativity index
• Isotope
• Hash function