Completing Partial Transversals of Cayley Tables of Abelian Groups

Jaromy Kuhl,Donald Mcginn,Michael William Schroeder

doi:10.37236/9386

Abstract

In 2003 Grüttmüller proved that if $n\geqslant 3$ is odd, then a partial transversal of the Cayley table of $\mathbb{Z}_n$ with length $2$ is completable to a transversal. Additionally, he conjectured that a partial transversal of the Cayley table of $\mathbb{Z}_n$ with length $k$ is completable to a transversal if and only if $n$ is odd and either $n \in \{k, k + 1\}$ or $n \geqslant 3k - 1$. Cavenagh, Hämäläinen, and Nelson (in 2009) showed the conjecture is true when $k = 3$ and $n$ is prime. In this paper, we prove Grüttmüller’s conjecture for $k = 2$ and $k = 3$ by establishing a more general result for Cayley tables of Abelian groups of odd order.

Highlights

It is well-known that Cayley tables of finite cyclic groups have transversals if and only if their order is odd
Observe that for any Abelian group G of odd order n, a partial transversal of C(G) with length 1, n − 1, or n is completable for the same reasons as when G is cyclic and for the same reasons, an appropriate first step in this generalization is to establish the conditions under which partial transversals of C(G) with lengths 2 and 3 are completable
We considered partial transversals in Cayley tables of Abelian groups of odd order, and we achieved our result through an iterative approach which may appear to only leverage that the group is solvable

Summary

Introduction

It is well-known that Cayley tables of finite cyclic groups have transversals if and only if their order is odd. For an Abelian group G of odd order n and k ∈ {2, 3}, every partial transversal of C(G) with length k is completable if and only if either n 3k − 1 or n ∈ {k, k + 1}. Observe that for any Abelian group G of odd order n, a partial transversal of C(G) with length 1, n − 1, or n is completable for the same reasons as when G is cyclic and for the same reasons, an appropriate first step in this generalization is to establish the conditions under which partial transversals of C(G) with lengths 2 and 3 are completable. There is a one-to-one correspondence between diagonally cyclic Latin squares of order n and transversals in C(Zn) as shown in the following example. For more on diagonally cyclic Latin squares and related objects, see the survey by Wanless [6]

Coset Blocks and Chains

Completing Partial Transversals

Concluding Remarks