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

Kyle Bonetta-Martin, Thomas A. McCourt

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 Zpn ⊕ (⨁ i = 1kZpai), where 1 ≤ k, 2 ≤ a1, a2, …, ak, p and n ≤ 1 + 2∑ i = 1k(ai − 1); and

  • with two exceptions and two potential exceptions any abelian group of rank two.


Keywords


Spherical latin bitrade, canonical group, abelian sand-pile group

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ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications