Latin bitrades, dissections of equilateral triangles, and abelian groups



Let T=(T*, T) be a spherical latin bitrade. With each a=(a1, a2, a3)∈T* associate a set of linear equations Eq(T, a) of the form b1+b2=b3, where b=(b1, b2, b3) runs through T*\{a}. Assume a1=0=a2 and a3=1. Then Eq(T,a) has in rational numbers a unique solution equation image. Suppose that equation image for all b, cT* such that equation image and i∈{1, 2, 3}. We prove that then T can be interpreted as a dissection of an equilateral triangle. We also consider group modifications of latin bitrades and show that the methods for generating the dissections can be used for a proof that T* can be embedded into the operational table of a finite abelian group, for every spherical latin bitrade T. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 1–24, 2010