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

TL;DR

This paper investigates which finite abelian groups can be realized as the canonical groups of spherical latin bitrades, proving some cannot and providing families that can, using connections to abelian sandpile groups.

Contribution

It establishes the existence of infinite families of abelian groups that do and do not arise as canonical groups of spherical latin bitrades, expanding understanding of their algebraic structure.

Findings

Some finite abelian groups do not arise as canonical groups.

Certain families of abelian groups are shown to be realizable as canonical groups.

The paper connects canonical groups to abelian sandpile groups of digraphs.

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 $\mathbb{Z}^{n}_p\oplus\left(\bigoplus_{i=1}^k\mathbb{Z}_{pa_i}\right)$, where $1\leq k$, $2\leq a_1,a_2,\ldots, a_k,p$ and $n\leq 1+2\sum_{i=1}^k(a_i - 1)$; and with one exception and three potential exceptions any abelian group of rank two.