Abelian-by-cyclic Moufang loops

TL;DR

This paper constructs new examples of finite abelian-by-cyclic Moufang loops using groups with triality, expanding known classes and demonstrating embeddability into Cayley algebras.

Contribution

It introduces a new series of nonassociative Moufang loops with abelian normal subloops and cyclic quotients, generalizing previous odd-order examples.

Findings

New series of finite abelian-by-cyclic Moufang loops

Some loops can be embedded into Cayley algebras

Generalization of previously known odd-order loops

Abstract

We use groups with triality to construct a series of nonassociative Moufang loops. Certain members of this series contain an abelian normal subloop with the corresponding quotient being a cyclic group. In particular, we give a new series of examples of finite abelian-by-cyclic Moufang loops. The previously known [A. Rajah, J. Alg., 235 (2001), 66-93] loops of this type of odd order 3q^3, with prime q congruent to 1 mod 3, are particular cases of our series. Some of the examples are shown to be embeddable into a Cayley algebra.