The Prime Graphs of Groups With Arithmetically 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.