无限群的CI问题
The CI problem for infinite groups
摘要：
一个有限群$G$是DCI-群，如果当$S$和$S'$是$G$的子集且Cayley图Cay $(G,S)$和Cay $(G,S')$同构时，存在$G$与$φ(S)=S'$的自同构$φ$。它是CI-群，如果在$S=S^{-1}$ .我们将这些定义推广到无限群，并给出了两个密切相关的定义：一个无限群是强（D）CI $_f$ -群，如果在$S$是有限的限制性假设下，相同的条件成立;一个无限群是（D）CI $_f$ -群，如果当$S$是有限的且生成$G$时，相同的条件成立。证明了无限（D）CI-群必是非局部有限的挠群。我们找到了（D）CI $_f$ -群但不是强（D）CI $_f$-群，以及强（D）CI $_f$ -群但不是（D）CI-群的无限群族.我们讨论这些属性中的哪些是由子群继承的。最后，我们完全刻划了$\mathbb Z^n$上的局部有限DCI-图.我们提出了几个与这些思想有关的公开问题，包括是否存在任何无限（D）CI-群的问题。
英文摘要：
A finite group $G$ is a DCI-group if, whenever $S$ and $S'$ are subsets of $G$ with the Cayley graphs Cay$(G,S)$ and Cay$(G,S')$ isomorphic, there exists an automorphism $φ$ of $G$ with $φ(S)=S'$. It is a CI-group if this condition holds under the restricted assumption that $S=S^{-1}$. We extend these definitions to infinite groups, and make two closely-related definitions: an infinite group is a strongly (D)CI$_f$-group if the same condition holds under the restricted assumption that $S$ is finite; and an infinite group is a (D)CI$_f$-group if the same condition holds whenever $S$ is both finite and generates $G$. We prove that an infinite (D)CI-group must be a torsion group that is not locally-finite. We find infinite families of groups that are (D)CI$_f$-groups but not strongly (D)CI$_f$-groups, and that are strongly (D)CI$_f$-groups but not (D)CI-groups. We discuss which of these properties are inherited by subgroups. Finally, we completely characterise the locally-finite DCI-graphs on $\mathbb Z^n$. We suggest several open problems related to these ideas, including the question of whether or not any infinite (D)CI-group exists.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
