Latin bitrades, dissections of equilateral triangles and abelian groups

2009/07/10 by Ales Drapal, Drapal, Ales, Carlo Hamalainen +3
Mathematics · #05B15 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05B15

paper · pdf · doi:10.48550/arxiv.0907.1789

arxiv created 2009/07/10 · arxiv updated 2009/12/01

Abstract

Let T = (T^{\textstyle ∗}, T^{\scriptscriptstyle \triangle}) be a spherical latin bitrade. With each a=(a₁,a₂,a₃)∈ T^{\textstyle ∗} associate a set of linear equations \eq(T,a) of the form b₁+b₂=b₃, where b = (b₁,b₂,b₃) runs through T^{\textstyle ∗} ∖ \a\. Assume a₁ = 0 = a₂ and a₃ = 1. Then \eq(T,a) has in rational numbers a unique solution b_i = b_i. Suppose that b_i ≠ c_i for all b,c ∈ T^{\textstyle ∗} such that b_i ≠ c_i and i ∈ \1,2,3\. We prove that then T^{\scriptscriptstyle \triangle} 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^{\textstyle ∗} can be embedded into the operational table of a finite abelian group, for every spherical latin bitrade T.

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• On which groups can arise as the canonical group of a spherical latin bitrade