A classification of prime graphs of pseudo-solvable groups

Abstract

The prime graph $\Gamma (G)$ 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 $\Gamma (G)$ 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 $\overline{\Gamma }(G)$ or $\{p,3,5\}$ form a triangle for some prime $p\ne 2$.