Uniqueness and explicit computation of mates in near-factorizations

TL;DR

This paper proves the uniqueness of mates in near-factorizations of finite groups, provides an explicit formula for their computation, and explores structural properties and existence in various group classes, including computational searches.

Contribution

It introduces an explicit formula for computing mates in near-factorizations and proves their uniqueness, offering new insights and methods in the study of group factorizations.

Findings

Mates in near-factorizations are unique.

An explicit formula enables efficient computation of mates.

Noncyclic abelian groups of order less than 200 lack nontrivial near-factorizations.

Abstract

We show that a "mate'' $B$ of a set $A$ in a near-factorization $(A,B)$ of a finite group $G$ is unique. Further, we describe how to compute the mate $B$ very efficiently using an explicit formula for $B$. We use this approach to give an alternate proof of a theorem of Wu, Yang and Feng, which states that a strong circular external difference family cannot have more than two sets. We prove some new structural properties of near-factorizations in certain classes of groups. Then we examine all the noncyclic abelian groups of order less than $200$ in a search for a possible nontrivial near-factorization. All of these possibilities are ruled out, either by theoretical criteria or by exhaustive computer searches. (In contrast, near-factorizations in cyclic or dihedral groups are known to exist by previous results.) We also look briefly at nontrivial near-factorizations of index $\lambda > 1$ in noncyclic abelian groups. Various examples are found with $\lambda = 2$ by computer.