2024/11/24 by Kreher, Donald L., Martin, William J., Stinson, Douglas R. · 1 citation
#05B10 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2411.15890
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.