测地线局部可测的群
Groups whose geodesics are locally testable
摘要：
一个正则的词集合是（$k$ -）局部可测试的，如果集合中一个词的成员资格由它的某个有界长度$k$的子词的性质决定。在这篇文章中，我们研究的所有测地词的集合关于某些生成集是（$k$ -）局部可测试的组，我们称这样的组（$k$ -）局部可测试的。我们证明了一个群是\klt{1}当且仅当它是自由阿贝尔群。证明了（$k$ -）局部可测群类在取有限直积下是闭的.我们还证明了一个局部可测群有2个以上的扭元共轭类。我们的工作涉及计算机调查的特定群体，为此目的，我们实施了一个算法在\GAP \计算一个有限状态自动机的语言等于一组的所有测地线（假设这种语言是正规的），从shortlex自动结构。我们提供了该算法的简要说明。
英文摘要：
A regular set of words is ($k$-)locally testable if membership of a word in the set is determined by the nature of its subwords of some bounded length $k$. In this article we study groups for which the set of all geodesic words with respect to some generating set is ($k$-)locally testable, and we call such groups ($k$-)locally testable. We show that a group is \klt{1} if and only if it is free abelian. We show that the class of ($k$-)locally testable groups is closed under taking finite direct products. We show also that a locally testable group has finitely many conjugacy classes of torsion elements. Our work involved computer investigations of specific groups, for which purpose we implemented an algorithm in \GAP\ to compute a finite state automaton with language equal to the set of all geodesics of a group (assuming that this language is regular), starting from a shortlex automatic structure. We provide a brief description of that algorithm.
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。
