论文标题
具有算术小组组成因子的组的主要图
The Prime Graphs of Groups With Arithmetically Small Composition Factors
论文作者
论文摘要
我们继续研究有限基团的主要图，也称为Gruenberg-Kegel图。有限组的主要图的顶点是组订单的主要除数，而当且仅当该组中有订单$ pq $的元素时，边缘连接了两个顶点$ p $和$ q $。仅以图形理论术语表征了可解决的组的主要图，就像唯一不可忽略的组成因子为$ A_5 $的组的主要图一样。在本文中，我们对所有组成因子具有算术量很小的组的主要图表进行了分类，即在其命令中不超过三个质数。我们发现所有此类图都有$ 3 $颜色的补充，并且我们基于该组的Nonabelian组成因子的确切类型和多重性提供了此类组的主要图表的完整特征。
We continue the study of prime graphs of finite groups, also known as Gruenberg-Kegel graphs. The vertices of the prime graph of a finite group are the prime divisors of the group order, and two vertices $p$ and $q$ are connected by an edge if and only if there is an element of order $pq$ in the group. Prime graphs of solvable groups have been characterized in graph theoretical terms only, as have been the prime graphs of groups whose only nonsolvable composition factor is $A_5$. In this paper we classify the prime graphs of all groups whose composition factors have arithmetically small orders, that is, have no more than three prime divisors in their orders. We find that all such graphs have $3$-colorable complements, and we provide full characterizations of the prime graphs of such groups based on the exact type and multiplicity of the nonabelian composition factors of the group.
