A classification of prime graphs of pseudo-solvable groups

TL;DR

This paper classifies the prime graphs of pseudo-solvable groups, extending previous work on solvable groups by characterizing graphs based on triangle conditions involving specific primes.

Contribution

It provides the first classification of prime graphs for pseudo-solvable groups, broadening understanding beyond solvable groups.

Findings

Prime graphs of pseudo-solvable groups are characterized by specific triangle conditions.

The classification involves the primes 2, 3, 5, and a prime p, with triangles in the complement graph.

This extends the known classification from solvable to 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 $p\neq 2$.