论文标题
不可溶解组的分类，其功率图是Cograph
Classification of non-solvable groups whose power graph is a cograph
论文作者
论文摘要
卡梅伦，曼纳和梅哈塔里调查了哪个有限组的问题，该问题允许一个动力图，它也称为cograph，也称为电报组（代数591（2022））。他们为尼尔植物组提供了分类，并为一般组提供了部分结果。但是，作者指出了针对分类的数字理论障碍。当假定组对PSL 2（Q）或SZ（Q）是同构并可能很难时，就会出现这些。在本文中，我们证明这些数字理论问题实际上是不可溶解的电源 - 画像群体分类的唯一障碍。具体而言，对于不可解决的情况，我们根据这些组对PSL 2（Q）或SZ（Q）的同构进行了电源 - 描绘组的分类。对于可解决的情况，我们能够精确地描述可解决的功率描述组的结构。我们获得了连接Gruenberg-Kegel图的可解决功率描述组的完整分类。此外，我们减少了Gruenberg-Kegel图与P-Group的分类分类的情况，该p组允许Prime Power Order的无定点自动形态学，这通常是一个开放的问题。
Cameron, Manna and Mehatari investigated the question of which finite groups admit a power graph that is a cograph, also called power-cograph groups (Journal of Algebra 591 (2022)). They give a classification for nilpotent groups and partial results for general groups. However, the authors point out number theoretic obstacles towards a classification. These arise when the groups are assumed to be isomorphic to PSL 2 (q) or Sz(q) and are likely to be hard. In this paper, we prove that these number theoretic problems are in fact the only obstacles to the classification of non-solvable power-cograph groups. Specifically, for the non-solvable case, we give a classification of power-cograph groups in terms of such groups isomorphic to PSL 2 (q) or Sz(q). For the solvable case, we are able to precisely describe the structure of solvable power-cograph groups. We obtain a complete classification of solvable power-cograph groups whose Gruenberg-Kegel graph is connected. Moreover, we reduce the case where the Gruenberg-Kegel graph is disconnected to the classification of p-groups admitting fixed-point-free automorphisms of prime power order, which is in general an open problem.
