A classification of the prime graphs of pseudo-solvable groups

Abstract

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.