Latin bitrades derived from groups

TL;DR

This paper explores how certain latin bitrades can be constructed from groups using their right translation action, revealing new minimal and homogeneous examples and connecting group properties with latin trade characteristics.

Contribution

It introduces a group-based method for deriving latin bitrades, enabling the construction of previously unknown minimal and homogeneous trades and linking group theory with combinatorial designs.

Findings

Constructed new latin bitrades from well-known groups.

Proved existence of minimal, k-homogeneous latin trades for all odd k ≥ 3.

Identified smallest known examples of such trades.

Abstract

A latin bitrade is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. Dr\'apal (\cite{Dr9}) showed that a latin bitrade is equivalent to three derangements whose product is the identity and whose cycles pairwise have at most one point in common. By letting a group act on itself by right translation, we show how some latin bitrades may be derived from groups without specifying an independent group action. Properties of latin trades such as homogeneousness, minimality (via thinness) and orthogonality may also be encoded succinctly within the group structure. We apply the construction to some well-known groups, constructing previously unknown latin bitrades. In particular, we show the existence of minimal, $k$-homogeneous latin trades for each odd $k\geq 3$. In some cases these are the smallest known such examples.