The prime graph Γ ⁢ ( G ) \Gamma(G) of a finite group 𝐺 (also known as the Gruenberg–Kegel graph) has as its vertices the prime divisors of | G | \lvert G\rvert , and p ⁢ - ⁢ q p\textup{-}q is an edge in Γ ⁢ ( G ) \Gamma(G) if and only if 𝐺 has an element of order p ⁢ q 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 the prime graphs of a more general class of groups we call pseudo-solvable. These are groups whose composition factors are either cyclic or isomorphic to A 5 A_{5} . The classification is based on two conditions: the vertices { 2 , 3 , 5 } \{2,3,5\} form a triangle in Γ ̄ ⁢ ( G ) \overline{\Gamma}(G) or { p , 3 , 5 } \{p,3,5\} form a triangle for some prime p ≠ 2 p\neq 2 . The ideas developed in this paper also lay the groundwork for future work on classifying and analyzing prime graphs of more general classes of finite groups.

中文翻译：

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伪可解群素图的分类

素数图 γ ⁢ （ G ） \伽玛(G) 有限群 𝐺（也称为 Gruenberg-Kegel 图）的顶点为 | G | \lvert G\rvert ， 和 p ⁢ - ⁢ q p\textup{-}q 是一个边 γ ⁢ （ G ） \伽玛(G) 当且仅当 𝐺 具有有序元素 p ⁢ q pq 。自 20 世纪 70 年代诞生以来，这些图表已被广泛研究。然而，对更大的群族可能的素图进行完全分类仍然是一个难题。对于可解群，这样的分类是在 2015 年发现的。在本文中，我们首次超越了可解群，并描述了我们称为伪可解群的更一般类别的素图。这些群的组成因子要么是循环的，要么是同构的 A 5 一个_{5} 。分类基于两个条件：顶点 { 2 , 3 , 5 } \{2,3,5\} 形成一个三角形 γ ̄ ⁢ （ G ） \overline{\伽玛}(G) 或者 { p , 3 , 5 } \{p,3,5\} 为一些质数形成一个三角形 p ≠ 2 p\neq 2 。本文提出的思想也为未来对更一般的有限群类的素图进行分类和分析的工作奠定了基础。