Latin Bitrades, Dissections of Equilateral Triangles, and Abelian Groups

Let T = (T*, T-Delta) he a spherical latin bitrade. With each a = (a(1), a(2), a(3)) is an element of 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 b(i) = (b) over bar (i). Suppose that (b) over bar (i) not equal (c) over bar (i) for all b, c is an element of T* such that (b) over bar (i) not equal c(i) and i is an element of {1, 2, 3}. We prove that then T-Delta 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. (C) 2009 Wiley Periodicals, Inc. J Combin Designs 18: 1-24, 2010