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When Is a Latin Square Based on a Group?

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Book cover Orthogonal Latin Squares Based on Groups

Part of the book series: Developments in Mathematics ((DEVM,volume 57))

Abstract

A question that presents itself to us is, given a Latin square, is it isotopic to the Cayley table of a group? Several approaches have been proposed for answering this question, dating back to the 1800s. Some of the answers that we will present will be based on configurations in the given square or given bordered square, the quadrangle criterion for Cayley tables of quasigroups in general and its variants for loops, and the Thomsen condition characterizing Latin squares based on abelian groups. We will present the rectangle rule, a criterion for the normal multiplication table of a loop to be based on a group; and we will present approaches that are based on permuting rows and/or columns of the given square, or that treat rows and/or columns of the given square as permutations. Our emphasis will be on particular properties of Latin squares that characterize Latin squares based on groups. We will not emphasize algorithmic solutions, or algebraic properties of loops or quasigroups that characterize loops and quasigroups that are isotopic to groups.

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Authors and Affiliations

  1. Mathematics and Statistics, Wright State University, Dayton, OH, USA

    Anthony B. Evans

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  1. Anthony B. Evans
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Evans, A.B. (2018). When Is a Latin Square Based on a Group?. In: Orthogonal Latin Squares Based on Groups. Developments in Mathematics, vol 57. Springer, Cham. https://doi.org/10.1007/978-3-319-94430-2_2

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