Isomorphism invariants for linear quasigroups

  • Jonathan D. H. SmithEmail author
  • Stefanie G. Wang


For a unital ring S, an S-linear quasigroup is a unital S-module, with automorphisms \(\rho \) and \(\lambda \) giving a (nonassociative) multiplication \(x\cdot y=x^\rho +y^\lambda \). If S is the field of complex numbers, then ordinary characters provide a complete linear isomorphism invariant for finite-dimensional S-linear quasigroups. Over other rings, it is an open problem to determine tractably computable isomorphism invariants. The paper investigates this isomorphism problem for \({\mathbb {Z}}\)-linear quasigroups. We consider the extent to which ordinary characters classify \({\mathbb {Z}}\)-linear quasigroups and their representations of the free group on two generators. We exhibit non-isomorphic \({\mathbb {Z}}\)-linear quasigroups with the same ordinary character. For a subclass of \({\mathbb {Z}}\)-linear quasigroups, equivalences of the corresponding ordinary representations are realized by permutational intertwinings. This leads to a new equivalence relation on \({\mathbb {Z}}\)-linear quasigroups, namely permutational similarity. Like the earlier concept of central isotopy, permutational similarity is intermediate between isomorphism and isotopy.


Quasigroup Isotopy Group isotope T-quasigroup Ordinary character Permutation similarity 

Mathematics Subject Classification




We are grateful to a referee for helpful comments on an earlier version of this paper.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


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Copyright information

© Instituto de Matemática e Estatística da Universidade de São Paulo 2019

Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA
  2. 2.Department of MathematicsTrinity CollegeHartfordUSA

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