Latin bitrades, dissections of equilateral triangles, and abelian groups

Let T =( T * , T ▵ ) be a spherical latin bitrade. With each a =( a 1 , a 2 , a 3 )∈ T * associate a set of linear equations Eq ( T, a ) of the form b 1 + b 2 = b 3 , where b =( b 1 , b 2 , b 3 ) runs through T * \{ a }. Assume a 1 =0= a 2 and a 3 =1. Then Eq ( T,a ) has in rational numbers a unique solution \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document} $b_{i}=\bar{b}_{i}$\end{document} . Suppose that \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document} $\bar{b}_{i}\not= \bar{c}_{i}$\end{document} for all b, c ∈ T * such that \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document} $\bar{b}_{i}\not= \bar{c}_{i}$\end{document} 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