群的同步性与弱完美图：综述

中文摘要 AI 辅助

同步的概念已从有限自动机，经由变换半群，被转移到置换群上：称有限集合 $\Omega$ 上的置换群 $G$ 是同步的，如果对于 $\Omega$ 上的任意非置换 $t$，变换幺半群 $\langle G,t\rangle$ 都包含一个秩为~$1$ 的元素。同步置换群是本原且基本的，因此（在 O'Nan--Scott 分类中）是仿射的、对角的或几乎单的。在早前的一篇论文中，作者与 João Araújo 和 Ben Steinberg 综述了当时关于同步置换群已知的结果。此后，利用“一个群是非同步的当且仅当它保持一个非平凡的弱完美图”这一事实，若干新的群类已被检验是否具有同步性质。所研究的案例包括具有多于两个 socle 因子的对角群、对称群的两种自然作用，以及典型群的某些作用；其证明分散在文献中。我的主要目的是将这些结果连同其证明汇集起来。作为第二个目标，我将解释背景，并评论一些相关主题。（阅读本文不需要了解早前那篇论文。）

英文摘要

The concept of synchronization has been transferred from finite automata, via transformation semigroups, to permutation groups: a permutation group $G$ on a finite set $Ω$ is said to be synchronizing if, for any non-permutation $t$ on $Ω$, the transformation monoid $\langle G,t\rangle$ contains an element of rank~$1$. Synchronizing permutation groups are primitive and basic, and so are affine, diagonal, or almost simple (in the O'Nan--Scott classification). In an earlier paper, the author, with João Araújo and Ben Steinberg, surveyed what was known at the time about synchronizing permutation groups. Since then, several new classes of groups have been tested for the synchronizing property, using the fact that a group is non-synchronizing if and only if it preserves a nontrivial weakly perfect graph. Cases studied include diagonal groups with more than two socle factors, two natural actions of symmetric groups, and some actions of classical groups; the proofs are scattered in the literature. My primary purpose is to collect these results with their proofs. As a second objective, I will explain the background, and comment on some related topics. (Reading this paper does not require knowledge of the earlier paper.)

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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