The Prime Graphs of Groups With Arithmetically Small Composition Factors

TL;DR

This paper classifies the prime graphs of finite groups with composition factors having at most three prime divisors, revealing they all have 3-colorable complements and providing detailed characterizations based on their composition factors.

Contribution

It offers a complete classification of prime graphs for groups with arithmetically small composition factors, extending previous characterizations to a broader class.

Findings

All such prime graphs have 3-colorable complements.

Characterizations depend on the type and multiplicity of nonabelian composition factors.

Provides a comprehensive classification for groups with small composition factors.

Abstract

We continue the study of prime graphs of finite groups, also known as Gruenberg-Kegel graphs. The vertices of the prime graph of a finite group are the prime divisors of the group order, and two vertices $p$ and $q$ are connected by an edge if and only if there is an element of order $pq$ in the group. Prime graphs of solvable groups have been characterized in graph theoretical terms only, as have been the prime graphs of groups whose only nonsolvable composition factor is $A_5$. In this paper we classify the prime graphs of all groups whose composition factors have arithmetically small orders, that is, have no more than three prime divisors in their orders. We find that all such graphs have $3$-colorable complements, and we provide full characterizations of the prime graphs of such groups based on the exact type and multiplicity of the nonabelian composition factors of the group.