The Heritage of Cayley-Sudoku Tables

TL;DR

This paper reviews and extends the theory of Cayley-Sudoku tables for finite groups, uncovering historical connections, introducing new constructions, and encouraging further exploration of their heritage.

Contribution

It uncovers the historical roots of certain constructions, introduces new instances inspired by Baer, and discusses recent reinventions of existing methods.

Findings

Uncovered heritage of Constructions 1 and 2 in earlier work.

Introduced new Cayley-Sudoku tables inspired by Baer.

Outlined recent reinventions of Construction 1.

Abstract

A Cayley-Sudoku table of a finite group G is a Cayley table for G subdivided into uniformly sized rectangular blocks, in such a way that each group element appears once in each block. They were introduced by J. Carmichael, K. Schloeman, and M. B. Ward [CSW], who also gave three ways to construct them. This paper has four aims. First, we review Constructions 1 and 2 of [CSW] and uncover their unexpected heritage in the work of R. Baer and J. Denes. Next we turn to some new instances of Construction 2 inspired by Baer, which answer an open question in [CSW]. Third, we provide a very brief outline of recent reinventions of special cases of Construction 1 in the literature. We conclude with an invitation to seek out the heritage of Construction 3. Portions of this paper appear in: Kady Hossner Boden and Michael B. Ward (2019) A New Class of Cayley-Sudoku Tables, Mathematics Magazine, 92:4, 243-251, DOI: 10.1080/0025570X.2019.1613949