标题： 真正非交换部分差集 显示英文标题

标题： Genuinely nonabelian partial difference sets

摘要： 强正则图（SRGs）在代数组合学领域提供了丰富的研究方向，融合了图论、线性代数、群论、有限域、有限几何和数论中的技术。 特别引人关注的是那些具有大自同构群的强正则图。 如果一个自同构群在图的顶点上作用规则（尖锐传递地），那么我们可以将该图与群的一个子集联系起来，即部分差分集（PDS），这使得我们能够应用群论的技术来研究该图。 过去四十年的研究主要集中在使用特征理论强大技术的阿贝尔PDS上。 然而，关于非阿贝尔PDS的工作却很少。 本文指出存在 \textit{genuinely nonabelian} 部分差分集，即对于参数集，其中非阿贝尔群是唯一可能的规则自同构群的部分差分集。 我们还介绍了证明特定参数集或特定SRG的阿贝尔PDS不可能的方法。 描述了四个无穷族的真正非阿贝尔PDS，其中两个——一个来自三角图，另一个来自由Godsil构造的完全图的Krein覆盖 \cite{Godsil_1992} ——是新的。 我们还介绍了通过计算机搜索发现的一个新的非阿贝尔PDS，并提出了未来一些可能的研究方向。 显示英文摘要

摘要： Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of \textit{genuinely nonabelian} PDSs, i.e., PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which -- one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil \cite{Godsil_1992} -- are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.