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

Abstract

A loop X is said to satisfy Moufang’s theorem if for every x, y, z∈ 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.