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arXiv 1504.01629math.GR
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本原群与同步

Primitive groups and synchronization

João Araújo, Wolfram Bentz, Peter J. Cameron, Gordon Royle, Artur Schaefer

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One broken link fixed; some changes on the ordering of the sections
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摘要：

令$Ω$是基数$n$的集合，$G$是$Ω$上的置换群，而$f:Ω\toΩ$是不是置换的映射。如果变换半群$\langle G,f\rangle$包含一个常数映射，我们称$G$ \emph{synchronizes} $f$;如果$G$是\emph{every}非置换，我们称$G$是\emph{synchronizing group}。同步组必须是基本组，但也有不同步的本原群。每一个非同步的本原群至少不能同步一个一致变换（也就是说，其核具有相等大小的部分的变换），并且先前已经证明，本原群同步每一个非一致变换。本文的第一个目标是证明这个猜想是假的，通过展示本原群，无法同步特定的非均匀变换的秩$5$和$6$。此外，我们产生的图的自同构群有大约$\sqrt{n}$ \emph{non-synchronizing ranks}，从而驳斥了另一个猜想的数量上的非同步职级的本原群。本文的第二个目标是扩展频谱的秩，它是已知的，本原群同步的每一个非均匀变换的秩。以前已经证明，度为$n$的本原群可表示秩为$n-1$和$n-2$的每个非均匀变换，这里将其推广到$n-3$和$n-4$。确定确切的频谱秩存在不同步的一些原始组的非均匀变换只是几个自然的，但可能是困难的，问题的自动机，本原群，图形和计算代数所产生的这项工作，这些都概述了在最后一节。

英文摘要：

Let $Ω$ be a set of cardinality $n$, $G$ a permutation group on $Ω$, and $f:Ω\toΩ$ a map which is not a permutation. We say that $G$ \emph{synchronizes} $f$ if the transformation semigroup $\langle G,f\rangle$ contains a constant map, and that $G$ is a \emph{synchronizing group} if $G$ synchronizes \emph{every} non-permutation. A synchronizing group is necessarily primitive, but there are primitive groups that are not synchronizing. Every non-synchronizing primitive group fails to synchronize at least one uniform transformation (that is, transformation whose kernel has parts of equal size), and it has previously been conjectured that a primitive group synchronizes every non-uniform transformation. The first goal of this paper is to prove that this conjecture is false, by exhibiting primitive groups that fail to synchronize specific non-uniform transformations of ranks $5$ and $6$. In addition we produce graphs whose automorphism groups have approximately $\sqrt{n}$ \emph{non-synchronizing ranks}, thus refuting another conjecture on the number of non-synchronizing ranks of a primitive group. The second goal of this paper is to extend the spectrum of ranks for which it is known that primitive groups synchronize every non-uniform transformation of that rank. It has previously been shown that a primitive group of degree $n$ synchronizes every non-uniform transformation of rank $n-1$ and $n-2$, and here this is extended to $n-3$ and $n-4$. Determining the exact spectrum of ranks for which there exist non-uniform transformations not synchronized by some primitive group is just one of several natural, but possibly difficult, problems on automata, primitive groups, graphs and computational algebra arising from this work; these are outlined in the final section.

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。

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