Uniqueness and explicit computation of mates in near-factorizations

Document Type

Article

Publication Date

5-15-2026

Department

Department of Mathematical Sciences

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 λ>1 in noncyclic abelian groups. Various examples are found with λ=2 by computer.

Publication Title

Journal of Algebraic Combinatorics

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