Transfer of difference structures: a new semidirect product framework
Abstract
Difference sets, external difference families and near-factorizations of groups are much-studied structures satisfying certain uniformity conditions on the differences or sums within or between elements of sets in groups. Research has centred on abelian groups, though there are classic results (such as Dillon's Dihedral Trick for difference sets and the work of Pêcher on near-factorizations) connecting abelian and non-abelian structures, which have recently regained attention. We present a new explicit framework enabling transfer of difference structures between groups using a semidirect product approach, establishing new tools and constructions, encompassing various previous results and addressing an open problem of Swartz et al. Motivated by the classic dihedral results, we focus on semidirect products G rtimes ℤ2. Transfer from abelian to non-abelian groups, and between distinct non-abelian groups, are both possible, and our framework can handle λ-fold near-factorizations, difference sets and external difference structures with arbitrarily many sets. We obtain new near-factorizations and difference structures in a range of groups, and can transfer various examples not transferrable by previous approaches. We showcase our approach by establishing a new infinite family of three-set abelian circular external difference families, then producing from this the first infinite family of non-abelian circular external difference families.
Related
- A Uniform Product-Difference Theorem for Dense Subsets of ℤ2
- Classification of analytic conformal product structures
- Product structures and decompositions of ℤ2×ℤ2-symmetric spaces
- Continuous family of compact quantum metric space structures from cocycle twisted crossed product C∗-algebras
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BibTeX
@misc{huczynska2026transfer,
title = {{Transfer of difference structures: a new semidirect product framework}},
author = {Sophie Huczynska and Struan McCartney and Carys Williams},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18897},
eprint = {2609.18897},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18897},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Transfer of difference structures: a new semidirect product framework
AU - Sophie Huczynska
AU - Struan McCartney
AU - Carys Williams
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18897
UR - https://arxiv.org/abs/2609.18897
AB - Difference sets, external difference families and near-factorizations of groups are much-studied structures satisfying certain uniformity conditions on the differences or sums within or between elements of sets in groups. Research has centred on abelian groups, though there are classic results (such as Dillon's Dihedral Trick for difference sets and the work of Pêcher on near-factorizations) connecting abelian and non-abelian structures, which have recently regained attention. We present a new explicit framework enabling transfer of difference structures between groups using a semidirect product approach, establishing new tools and constructions, encompassing various previous results and addressing an open problem of Swartz et al. Motivated by the classic dihedral results, we focus on semidirect products G rtimes ℤ_2. Transfer from abelian to non-abelian groups, and between distinct non-abelian groups, are both possible, and our framework can handle λ-fold near-factorizations, difference sets and external difference structures with arbitrarily many sets. We obtain new near-factorizations and difference structures in a range of groups, and can transfer various examples not transferrable by previous approaches. We showcase our approach by establishing a new infinite family of three-set abelian circular external difference families, then producing from this the first infinite family of non-abelian circular external difference families.
ER -