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

  • Aleš Drápal
  • Petr VojtěchovskýEmail author


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.


Moufang’s theorem Steiner loop Steiner triple system Pasch configuration 

Mathematics Subject Classification

20N05 05B07 



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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsCharles UniversityPraha 8Czech Republic
  2. 2.Department of MathematicsUniversity of DenverDenverUSA

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