Uniqueness and explicit computation of mates in near-factorizations
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| Format: | Preprint |
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2024
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| _version_ | 1866929603733356544 |
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| author | Kreher, Donald L. Martin, William J. Stinson, Douglas R. |
| author_facet | Kreher, Donald L. Martin, William J. Stinson, Douglas R. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_15890 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniqueness and explicit computation of mates in near-factorizations Kreher, Donald L. Martin, William J. Stinson, Douglas R. Group Theory Combinatorics 05B10 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. |
| title | Uniqueness and explicit computation of mates in near-factorizations |
| topic | Group Theory Combinatorics 05B10 |
| url | https://arxiv.org/abs/2411.15890 |