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 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.

Keywords

• variety
• Steiner
• loops
• satisfying
• Moufangs
• theorem
• solution
• Rajahs
• Problem