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 $b_{i}=\bar{b}_{i}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 $b_{i}=\bar{b}_{i}$. Suppose that $\bar{b}_{i}\not= \bar{c}_{i}$ for all b, c∈T^* such that $\bar{b}_{i}\not= \bar{c}_{i}$ 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