论文标题
伪可溶剂组的主要图的分类
A classification of prime graphs of pseudo-solvable groups
论文作者
论文摘要
有限的组$ g $（也称为Gruenberg-Kegel图）的Prime Graph $γ（g）$具有$ | g | $的素数为顶点，而$ p \ text-q $是$γ（g）$的优势，并且仅当$ g $ g $具有订单$ pq $的元素。自1970年代成立以来，这些图表已经进行了广泛的研究。但是，对较大的群体家庭完全对可能的主要图进行了分类仍然是一个困难的问题。对于可解决的组，在2015年发现了这样的分类。在本文中，我们首次超越了可解决的组，并表征了我们称为伪可溶性的更通用组的素数。这些是其组成因子是循环或$ A_5 $的组。该分类基于两个条件：顶点$ \ {2,3,5 \} $在$ \overlineγ（g）$或$ \ {p，3,5 \} $中形成三角形的三角形，形成了一些prime $ p \ neq 2 $的三角形。
The prime graph $Γ(G)$ of a finite group $G$ (also known as the Gruenberg-Kegel graph) has as its vertices the prime divisors of $|G|$, and $p\text-q$ is an edge in $Γ(G)$ if and only if $G$ has an element of order $pq$. Since their inception in the 1970s these graphs have been studied extensively; however, completely classifying the possible prime graphs for larger families of groups remains a difficult problem. For solvable groups such a classification was found in 2015. In this paper we go beyond solvable groups for the first time and characterize prime graphs of a more general class of groups we call pseudo-solvable. These are groups whose composition factors are either cyclic or $A_5$. The classification is based on two conditions: the vertices $\{2,3,5\}$ form a triangle in $\overlineΓ(G)$ or $\{p,3,5\}$ form a triangle for some prime $p\neq 2$.
