A variety of Steiner loops satisfying Moufang's theorem: A solution to Rajah's Problem

TL;DR

This paper identifies a new class of Steiner loops that satisfy Moufang's theorem but are not Moufang loops themselves, solving a problem posed by Andrew Rajah.

Contribution

It introduces a specific variety of Steiner loops satisfying a unique identity, expanding understanding of loops related to Moufang's theorem.

Findings

The variety of Steiner loops satisfying the given identity is not contained in Moufang loops.

Every loop in this variety satisfies Moufang's theorem.

The paper provides a solution to Rajah's problem.

Abstract

A loop $X$ is said to satisfy Moufang's theorem if for every $x,y,z\in X$ such that $x(yz)=(xy)z$ the subloop generated by $x$, $y$, $z$ is a group. We prove that the variety $V$ of Steiner loops satisfying the identity $(xz)(((xy)z)(yz)) = ((xz)((xy)z))(yz)$ is not contained in the variety of Moufang loops, yet every loop in $V$ satisfies Moufang's theorem. This solves a problem posed by Andrew Rajah.