The Prime Graphs of Some Classes of Finite Groups
The Prime Graphs of Some Classes of Finite Groups
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DOI：
10.1016/j.jpaa.2021.106990
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发表时间：
2021-01
影响因子：
0.8
通讯作者：
Chris Florez;Jonathan Higgins;Kyle Huang;T. Keller;Dawei Shen;Yong Yang
中科院分区：
文献类型：
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作者：
Chris Florez;Jonathan Higgins;Kyle Huang;T. Keller;Dawei Shen;Yong Yang
在本文中，我们研究有限群的素图。有限群G的素图，也称为Gruenberg-Kegel图，是顶点集{素数除||的图。 G|} 和边 p-q 当且仅当 G 中存在 pq 阶元素。在有限群论中，研究群的素图一直是过去近半个世纪的重要课题。直到最近，可解群的素图才仅用图论术语来表征。在本文中，我们继续这一研究方向，并给出了几类群的完整表征，包括无平方阶群、元幂群、无立方阶群，以及对于任何 n∈ N，n 次方无阶群的可解群。我们还探索了组成因子为循环或 A 5 的群的素图，并与 Maslova 的猜想建立了联系。然后，我们提出了一种从对偶素数图恢复素数图的算法。
In this paper we study prime graphs of finite groups. The prime graph of a finite group G, also known as the Gruenberg-Kegel graph, is the graph with vertex set {primes dividing| G|} and an edge p-q if and only if there exists an element of order pq in G. In finite group theory, studying the prime graph of a group has been an important topic for the past almost half century. Only recently prime graphs of solvable groups have been characterized in graph theoretical terms only. In this paper, we continue this line of research and give complete characterizations of several classes of groups, including groups of square-free order, metanilpotent groups, groups of cube-free order, and, for any n∈ N, solvable groups of n th-power-free order. We also explore the prime graphs of groups whose composition factors are cyclic or A 5 and draw connections to a conjecture of Maslova. We then propose an algorithm that recovers the prime graph from a dual prime graph.