On abelian-by-cyclic Moufang loops

TL;DR

This paper classifies and constructs abelian-by-cyclic Moufang loops using special permutations called Moufang permutations and introduces construction pairs, revealing conditions under which these loops are groups.

Contribution

It provides a complete construction method for split 3-divisible abelian-by-cyclic Moufang loops using Moufang permutations and extends this with construction pairs.

Findings

All split 3-divisible abelian-by-cyclic Moufang loops can be constructed from Moufang permutations.

The abelian subgroup induces an abelian congruence only when the loop is a group.

New classes of abelian-by-cyclic Moufang loops are obtained via construction pairs.

Abstract

We study abelian-by-cyclic Moufang loops. We construct all split $3$-divisible abelian-by-cyclic Moufang loops from so-called Moufang permutations on abelian groups $(X,+)$, which are permutations that deviate from an automorphism of $(X,+)$ by an alternating biadditive mapping (satisfying certain properties). More generally, we obtain additional abelian-by-cyclic Moufang loops from so-called construction pairs. As an aside, we show that in the Moufang loops $Q$ obtained from a construction pair on $(X,+)$ the abelian normal subgroup $(X,+)$ induces an abelian congruence of $Q$ if and only if $Q$ is a group.