差结构的转移：一个新的半直积框架

中文摘要 AI 辅助

差集、外部差族和群的近因子分解是研究较多的结构，它们满足群中集合内部或集合之间元素差或和上的某些均匀性条件。研究主要集中在阿贝尔群上，尽管有一些经典结果（如Dillon关于差集的二面体技巧和Pêcher关于近因子分解的工作）将阿贝尔与非阿贝尔结构联系起来，这些结果最近重新受到关注。我们提出了一个新的显式框架，通过半直积方法实现群之间差结构的转移，建立了新的工具和构造，涵盖了之前的多项结果，并解决了Swartz等人提出的一个开放问题。受经典二面体结果的启发，我们专注于半直积$G \rtimes \mathbb{Z}_2$。从阿贝尔群到非阿贝尔群的转移，以及不同非阿贝尔群之间的转移都是可行的，我们的框架可以处理具有任意多个集合的$\lambda$重近因子分解、差集和外部差结构。我们在多种群中获得了新的近因子分解和差结构，并且可以转移之前方法无法转移的各种例子。我们通过建立一个新的无限族三集合阿贝尔循环外部差族来展示我们的方法，并由此产生了第一个无限族非阿贝尔循环外部差族。

英文摘要

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 $G \rtimes \mathbb{Z}_2$. 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.

发表机构

• School of Mathematics and Statistics, University of St Andrews（圣安德鲁斯大学数学与统计学院）

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