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

2016/06/27 by Kyle Bonetta-Martin, Bonetta-Martin, Kyle, Thomas A. McCourt +1
Mathematics · #05B15 #05C10 #05C20 #05C25 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05B15 #msc:05C10 #msc:05C20 #msc:05C25

paper · pdf · doi:10.48550/arxiv.1606.08122

arxiv created 2016/06/27 · arxiv updated 2016/06/28

Abstract

We address a question of Cavenagh and Wanless asking: which finite abelian groups arise as the canonical group of a spherical latin bitrade? We prove the existence of an infinite family of finite abelian groups that do not arise as canonical groups of spherical latin bitrades. Using a connection between abelian sandpile groups of digraphs underlying directed Eulerian spherical embeddings, we go on to provide several, general, families of finite abelian groups that do arise as canonical groups. These families include: any abelian group in which each component of the Smith Normal Form has composite order; any abelian group with Smith Normal Form ℤⁿ_p⊕(\bigoplus_i=1^kℤ_pai), where 1≤ k, 2≤ a₁,a₂,…, a_k,p and n≤ 1+2∑_i=1^k(a_i - 1); and with one exception and three potential exceptions any abelian group of rank two.

Related

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• Latin bitrades, dissections of equilateral triangles and abelian groups
• Spherical space forms revisited
• Growth rates of groups associated with face 2-coloured triangulations and directed Eulerian digraphs on the sphere
• On symmetries of spheres in univalent foundations